The goal of this clinical research study is to learn if giving umbilical cord blood along with standard stem cells after high-dose chemotherapy will improve the response to a stem cell transplant. The safety of this treatment will also be studied.

Cord blood is a source of blood-forming cells that can be used for transplantation. Cord blood cells are taken from the umbilical cords of women who have given birth, and who have volunteered to donate their umbilical cord. Researchers hope that using cord blood before a stem cell transplant will help to reduce the risk of graft-versus-host-disease (GVHD). GVHD occurs when donor cells attack the cells of the person receiving the transplant.

If you are found to be eligible to take part in this study, you will receive chemotherapy for 5 days. You will receive fludarabine over about 30 minutes through a needle in your vein on Days 1-4. You will receive melphalan through a needle in your vein over about 30 minutes on Day 5.

On Day 7, you will receive the cord blood cells through a needle in your vein over about 30 minutes.

On Day 12, you will receive blood stem cells through a needle in your vein over 30-60 minutes. The stem cells you receive will be from a stem cell donor whose human leukocyte antigen (HLA- proteins on cells) type matches yours.

If appropriate for the disease, you will also receive rituximab about once weekly for 4 weeks, beginning on the day you receive melphalan. Rituximab is given though a needle in your vein over 2-3 hours.

You will receive the drugs tacrolimus and methotrexate to lower the risk of GVHD. Tacrolimus will be given through a needle in your vein non-stop for 2 weeks, starting 12 hours after the stem cell transplant. After the first 2 weeks, you will continue to receive tacrolimus by mouth, for at least 3 months. You will receive methotrexate though a needle in your vein over 30 minutes, starting 1 day after the stem cell transplant, for a total of 3 doses over the first 6 days after the stem cell transplant.

You will receive the G-CSF (granulocyte-colony stimulating factor) to help you blood cell counts recover. G-CSF will be given as an injection under the skin, beginning 1 week after the stem cell transplant. You will continue to receive G-CSF once a day until your blood cell counts reach a certain high enough level.

You will need to stay in the hospital for about 4 weeks beginning on Day 1. While you are in the hospital, blood (about 2 teaspoons) will be drawn every day for routine tests.

After you leave the hospital, you will return to the hospital for visits 2-3 times a week for at least 100 days after the transplant. During these visits, you will have a physical exam and blood (about 2 teaspoons) will be drawn for routine tests.

You will be asked to come back to the clinic for follow-up visits at 3, 6, 9 and 12 months after your transplant for routine safety testing. This will include a physical exam, a bone marrow biopsy, and blood (about 2 teaspoons) will be drawn for routine testing.

You will be considered off-study after the 12-month follow-up visit.

This is an investigational study. The stem cell transplant, the umbilical cord transplant, and all drugs used on this study are FDA approved. The use of umbilical cord blood and stem cells together is investigational. Up to 30 patients will take part in this study. All will be enrolled at M. D. Anderson.

Inclusion Criteria:

1. Patients with multiple myeloma (MM), acute myeloid leukemia (AML), myelodysplastic syndrome (MDS), acute lymphoblastic leukemia (ALL), non-Hodgkin's lymphoma (NHL), Hodgkin's lymphoma (HL), or chronic lymphocytic leukemia (CLL) in greater than first complete remission who are candidates for a non-ablative or reduced intensity conditioning regimen.
2. Age up to 80 years.
3. A related or unrelated donor who is HLA-matched at HLA, A, B, C, DR and DQ loci is acceptable (i.e. 10/10 matched related or unrelated donor, matched with molecular high-resolution technique per current standard for the BMT program). Donor must be willing to donate peripheral blood or bone marrow progenitor cells.
4. Available cord blood unit must contain a minimum of 1.5 * 10^7 total nucleated cells per kg, and be at least a 4/6 HLA match with patient.
5. Zubrod PS less than or equal to 2 or Lansky PS greater than or equal to 50%.
6. Left ventricular ejection fraction >40%. No uncontrolled arrhythmias or symptomatic heart disease.
7. Forced Expiratory Volume in 1 second (FEV1), Forced Vital Capacity (FVC) and Diffusion Capacity (DLCO) >40%.
8. Serum creatinine <2.0 mg/dL. Serum bilirubin <3 * upper limit of normal, SGPT <4 * upper limit of normal.

Exclusion Criteria:

1. Patients with active CNS disease
2. Positive Beta HCG in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization.
3. Serious medical or psychiatric illness likely to interfere with participation in this clinical study, including but not limited to active uncontrolled infection, uncontrolled cardiac arrhythmia or ischemic event, or uncontrolled psychosis, major depression, or mania.
4. Evidence of chronic, active hepatitis or cirrhosis, or HIV

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