The goal of this clinical research study is to learn if combining cord blood units to make the cells "take" faster in recipients will help to improve the results of cord blood transplants.

Cord blood is a source of blood forming cells that can be used for transplantation. The major problem with this type of transplant is the small number of blood forming cells available in each cord unit, which may delay the "take" of the graft in the recipient. Two strategies may be used to try to overcome this problem. One method is the combination of 2 units of cord blood and the other is the growing of the cord blood cells in the laboratory before they are transplanted in order to increase their number.

Participants will be randomly assigned (as in the toss of a coin) to one of two groups. If you are assigned to Study Arm 1, you will receive two cord blood units combined without growing the cells in the laboratory. If you are assigned to Study Arm 2, you will receive one cord blood unit combined with one cord blood unit which will be grown in the laboratory for two weeks before you receive it. Neither you nor your doctors will know in advance to which arm you will be assigned.

Placement of central venous catheter for collection of "back-up" stem cells:

Before you have back-up stem cell collected or you receive chemotherapy, you will require placement of a hollow plastic tube (catheter) into a large vein inside your body. This catheter will be used to draw blood and to give medications and fluids. The catheter is inserted through the skin in the upper chest and extends into the right side of your heart. Your doctor will explain this procedure to you in more detail, and you will be required to sign a separate consent form for it.

Collection of back up stem cells:

Because collecting additional cells from the donor of the cord blood will not be possible if the transplant with cord blood fails, a back up blood or bone marrow sample will be collected from you and frozen before the high dose chemotherapy begins.

Peripheral blood progenitor cell collection (Leukapheresis):

Before collection of the blood stem cells, you will be treated with a drug called granulocyte colony stimulating factor (G-CSF), which will cause the important stem cells in the marrow to move the peripheral blood where they will be collected. This medication is given as a shot under the skin once a day for 3 - 7 days, at which time your blood stem cells will be collected from your central catheter during a 3-4 hour out-patient procedure. In some cases where a "good vein" cannot be accessed, a silicone venous catheter may be needed to collect your blood.

Bone Marrow Collection:

If the leukapheresis cannot be performed successfully, you will receive general anesthesia in the operating room and will have multiple needle sticks of the hip bones in order to collect bone marrow. Only a small part of your bone marrow (<5%) will be taken.

Selection of another donor as the alternate source of stem cells:

If your bone marrow or back-up blood stem cells cannot be collected, a family member whose bone marrow is the closest match will be chosen as the back up donor. The potential donor would have testing performed to assure that he/she is eligible to donate.

High-dose therapy:

Your bone marrow transplant doctors will give you one of four chemotherapy treatments, which are discussed below. If you have acute lymphoblastic leukemia, lymphoma, Hodgkin's disease, chronic lymphocytic leukemia, acute myelogenous leukemia, myelodysplastic syndrome, chronic eosinophilic leukemia or chronic myelogenous leukemia, are less than 61 years old, and can receive high-dose chemotherapy, you may be assigned to Treatment Regimen #1 - fludarabine-thiotepa-melphalan. Patients with ALL may be assigned to Treatment Regimen #1 or #3, depending on your doctor's evaluation, age and physical fitness.

You will receive melphalan as a single dose on Day -8, thiotepa as a single dose on Day -7, followed by fludarabine given once a day for 4 days in a row (Days -6 through -3. Rituximab may be given on Day -9 if appropriate for your disease. Day 0 is the day of transplantation, so the negative day numbers are used to label the treatment days before transplant.

If you have any of the diseases listed above, have had a bone marrow transplant in the past, or have a physical that makes you less likely to tolerate high-dose therapy well, you will be in the less aggressive Treatment Regimen #2, Fludarabine, Cyclophosphamide and low dose total body irradiation. The fludarabine will be given on Days -6, -5, -4, and -3. The cyclophosphamide will be given on Day -6. Irradiation will be given on day -1. Rituxan may be given on Day -7 if appropriate for your disease.

If you have acute lymphoblastic leukemia, are 50 years old or younger, and your doctors have decided that full-dose Total body irradiation is the best treatment for you, you will be assigned to Treatment Regimen #3 - total body irradiation-VP16. You will receive total body irradiation on Days -7,-6, -5 and -4, followed by VP16 (also known as etoposide) as a single dose on Day -3. Rituximab may be given on Day -8 if appropriate for your disease.

All chemotherapy, fluids and other medications that must be given by vein will be infused through your catheter. Once the backup stem cells are collected, all patients will be admitted to the hospital on Day -9 to begin receiving fluids. Chemotherapy may be stopped if intolerable side effects occur.

Expansion of expanded cord blood:

On Day -14, if you are in Treatment Group 2, one of your two cord blood units will be thawed and treated in the MD Anderson Stem Cell Laboratory with vitamin-like growth factors for 2 weeks before they are given back to you on day zero as described below. A small amount of cord blood cells (less than 3%) will be used for laboratory procedures that measure the quality of the product.

The CliniMACS System is a medical device that is used to separate types of blood cells from blood that is removed from the body during leukapheresis. These separated cells are processed for use in treatments such as stem cell transplants.

Transplantation of cord blood:

If you are in Treatment Group 1, two days following completion of high-dose therapy (Day 0), both units of cord blood will be thawed and infused (one at a time) through your catheter. Each unit will take about 30 minutes to infuse. Patients in Treatment Group 2 will receive one unit of cord blood that did not go through expansion of cells, followed by infusion of the unit that was grown in the laboratory for 14 days.

Graft versus host disease (GVHD) preventive therapy:

GVHD results from a reaction of the transplanted cord blood cells against certain tissues in your body. In an attempt to prevent or decrease the severity of GVHD, you will receive two drugs.

Mycophenolate mophetyl (MMF) pills will be given starting three days before your transplant, and will be continued until day 30 after the cord blood transplant. If you cannot take pills, the drug can be given through your catheter. If you develop GVHD, the use of MMF may be prolonged.

Tacrolimus will be given as a 24 hour continuous infusion over 3-6 weeks. Around Day 30 or 40 (after engraftment), the tacrolimus will be changed to pills given daily for 6-9 months. The number of tacrolimus pills may vary according to the blood levels of the drug, but usually are between 1 - 5. This medicine is used for 6-9 months (longer if chronic GVHD occurs).

You will remain on study as long as your disease does not return. If your disease returns, you will be taken off study and you may be offered participation in another study or be treated outside of this study.

Follow-up after transplant:

After you leave the hospital, you will be seen regularly in the Department of Blood and Marrow Transplantation at MD Anderson. The frequency of the visits may vary, but may be as often as daily. Blood (1-2 tablespoons) and urine tests will be performed. The frequency of blood tests may also vary, but may be performed daily. Patients will need a bone marrow sample collected before transplant and at 1 month, 3-4 months, and 5-7 months after transplant. After that, bone marrow samples will be collected once a year, indefinitely. To collect a bone marrow sample, an area of the hip or chest bone is numbed with anaesthetic and a small amount of bone marrow is withdrawn through a large needle. Patients with lymphomas and Hodgkin's disease will need CT scan of the thorax, abdomen, and pelvis performed before transplant and then at 1 month, 3-4 months, and 5-7 months after the transplant. After that it will be done yearly.

This is an investigational study. All treatment drugs are FDA approved and commercially available.

A total of 110 patients may take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Disease-Specific Eligibility Requirements: Patients must have one of the following hematologic malignancies: 1. Acute Myelogenous Leukemia (AML), Myelodysplastic Syndrome (MDS) 2. Acute Lymphoblastic Leukemia (ALL) 3. Chronic Myelogenous Leukemia (CML) 4. Non-Hodgkin's Lymphoma (NHL) 5. Hodgkin's Disease (HD) 6. Chronic Lymphocytic Leukemia (CLL) 7. Chronic eosinophilic leukemia or Philadelphia chromosome negative CML.
2. Greater than 1 month old and <=60 years old for full myeloablative therapy.
3. Patients must have two CB units available which are matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 1E7 total nucleated cells/Kg recipient body weight in the pre-thawed fraction.
4. Patient must be willing to undergo bone marrow harvest or peripheral blood progenitor cell (PBPC) collection for use in case of engraftment failure. If the patient is unable or fails to successfully undergo the collection, a family member must be identified to donate hematopoietic stem cells for haploidentical transplant in case of engraftment failure. If autologous hematopoietic stem cells cannot be procured due to marrow contamination by malignancy, or due to harvest failure, and a haploidentical relative is not available or not willing to donate, two cord blood units can be used as the back-up graft.
5. Continuation to Criteria # 4: These units will be identified prior to enrollment in this study.
6. Regimen 1 (Myeloablative mel/thiotepa/fludarabine): 1.Patients with ALL, HD, NHL, AML, MDS, CML, CLL and Chronic eosinophilic leukemia who are candidates for full myeloablative therapy. 2.Performance score of at least 60% by Karnofsky (age >= 12 years), or Lansky Play-Performance Scale (age <12 years). 3.Age >=1 month <=60 years (high-dose).
7. Continuation to Criteria # 6: 4.Adequate major organ system function as demonstrated by:a. Left ventricular ejection function of at least 40%. b.Pulmonary function test demonstrating a diffusion capacity of at least 50%. predicted (high-dose). c.Creatinine < 1.6 mg/dL. d.serum glutamate pyruvate transaminase (SGPT)/bilirubin <= to 2.0 x normal (high-dose).
8. Eligibility for Regimen 2 (Non-myeloablative Cy-Flu-TBI): 1. Patients with ALL, AML, MDS, CML, NHL, CLL, Chronic eosinophilic leukemia and HD who are not candidates for full myeloablative therapy. All patients who received a prior autologous transplant are eligible. 2. Performance score of at least 60% by Karnofsky or PS < 3 (ECOG) (age >= 12 years), or Lansky Play-Performance Scale (age <12 years) 3. Age >= 1 month <=80 years
9. Continuation to Criteria # 8: 4. Left ventricular ejection function of at least 30%; 5. Pulmonary function test demonstrating a diffusion capacity of at least 40% predicted; 6. Creatinine < 3.0 mg/dL; 7. SGPT <= to 4.0 x normal.
10. Regimen 3 (Myeloablative VP16-TBI): 1. Patients with ALL who are candidates for myeloablative therapy, and require a TBI-containing regimen. 2. Performance score of at least 60% by Karnofsky or PS < 2 (ECOG) (age >= 12 years), or Lansky Play-Performance Scale (age <12 years). 3. Age >= 1 month <=50 years. 4. Organ function requirements: a. Left ventricular ejection function of at least 50%. b. Pulmonary function test demonstrating a diffusion capacity of at least 50% predicted. c. Creatinine < 1.6 mg/dL. d. SGPT <= 2.0 x normal.

Exclusion Criteria:

1. HIV positive.
2. Pregnancy.
3. Serious medical Condition.
4. Patients with signs & symptoms leading to positive lumbar puncture (malignant cells in the CSF) or to documented metastatic parenchymal disease are ineligible for this study.
5. Availability of appropriate, willing, HLA-matched related donor.