The goal of this clinical research study is to learn if an infusion of white blood cells (called T cells) that have been genetically changed is safe to give patients who have received an umbilical cord blood transplant (UCBT). Researchers want to learn if these genetically changed T-cells are effective in attacking cancer cells in patients with advanced B-cell lymphoma or leukemia after they have received an UCBT, how long the changed T-cells stay in the body, and if adding them to standard transplant could improve how patients respond to treatment.

Funding Source - FDA OOPD

T-cell infusion:

The gene transfer used in this study involves umbilical cord blood cells taken either from your UCB unit(s) or from cells left over after your UCB unit has been processed. Researchers perform a gene transfer to change the T-cells' DNA (the genetic material of cells), and then inject the changed T-cells into the body of the patient receiving the transplant. This process is called a modified donor lymphocyte infusion (DLI) or T-cell infusion.

Chemotherapy and UCBT:

After the umbilical cord blood unit has been identified for your transplant and after it has arrived at the hospital, you will be admitted to the hospital to receive chemotherapy and the UCBT. These procedures are not considered part of this research study. You will discuss these procedures with a study doctor and sign an informed consent document with specific details of the UCBT procedure and possible risks, at another time.

You will be given standard drugs to help decrease the risk of side effects. You may ask the study staff for information about how the drugs are given and their risks.

The T-cell Infusions:

After the UCBT, the study doctor will decide when you will be eligible for the T-cell infusions. You must be at least 42 days past your UCBT without any serious evidence of active graft versus host disease (GVHD). GVHD occurs when the donor cells attack the cells of the person receiving the transplant.

The T-cell infusion is given by vein, usually over 15-30 minutes over 2 days. A smaller dose of T cells will be infused on the first day and if this is tolerated the remaining dose of T cells will be infused within 72 hours after the first infusion. During both infusions, your vital signs will be checked.

Study Tests:

Within 30 days before the T-cell infusion:

• Blood (about 2 tablespoons) will be drawn to check how well the donated blood cells and tissue are mixing with your blood. Researchers use this test to find out how successful the transplant is. This is a standard test for patients who have received a transplant.
• Blood (about 2 tablespoons) will be drawn for research tests to measure the number of B-cells and other (non-transplanted) T-cells.

Within 7 days before the T-cell infusion:

• You will have a physical exam.
• You will be checked for possible reactions to your UCBT, including GVHD and graft failure.
• Blood (about 4 tablespoons) will be drawn for routine tests, to check your immune system, and to check for HAMA immune system reactions.

Within 24 hours, and then about 3 days, 1 week (+/- 2 days), 2 weeks (+/- 2 days), 4 weeks (+/- 7 days), 8 weeks (+/- 7 days), 3 months (+/- 7 days), 6 months (+/- 7 days), and 12 months (+/- 7 days) after the T-cell infusion:

• You will have a physical exam.
• You will be checked for possible reactions to your treatment, including GVHD and graft failure.
• Blood (about 4 tablespoons) will be drawn for routine tests.

About 1 month (+/- 7 days), 2 months (+/- 7 days), 3 months (+/- 7 days), 6 months (+/- 7 days), and 12 months (+/- 7 days) after the T-cell infusion, blood (about 2 tablespoons) will be drawn to check certain protein levels. This may be checked more often, and the protein may be replaced, if your doctor thinks it is needed.

Once during Weeks 2 and 4, and then once a month for 6 months after the T-cell infusion, blood (about 1 teaspoon) will be drawn to check your immune system. This blood draw may be done at the same time as the blood draws for routine tests when possible, to avoid extra "needle sticks".

After the 2nd T-cell infusion:

• Blood (about 2 tablespoons) will be drawn for research tests to check the level of the infused T-cells and to measure the number of B-cells and other (non-transplanted) T-cells.

About 30 days and 3, 6, and 12 months after the T-cell infusion:

• You will have a physical exam, and you will be checked for possible reactions to your treatment, including GVHD and graft failure.
• Blood (about 2 tablespoons) will be drawn for research tests to check the level of the infused T-cells and to measure the number of B-cells and other (non-transplanted) T-cells. °During the Month 3 visit, part of this blood sample will be used to check for HAMA immune system reactions.
• Urine will be collected for routine tests.
• You will have a bone marrow aspiration and biopsy performed to check the status of the disease.
• If you have lymphoma, you may have a you will have a positron emission tomography - computed tomography (PET-CT) scan and/or a CT scan of your neck, chest, abdomen, and pelvis to check the status of the disease, if your doctor thinks it is needed.

Tests and/or procedures may be repeated more often, if your study doctor thinks it is needed.

Length of Study:

You may continue taking part in this study for up to 12 months. You will be taken off study if the disease gets worse, you have any infections, intolerable side effects occur, you are not able to follow study directions, the T-cells do not engraft (grow in your body), or you are not able to receive a T-cell infusion.

You should talk to the study doctor if you want to leave the study early. If you are taken off study early, you still may need to return for routine post-transplant follow-up visits, or if your transplant doctor decides it is needed.

Your participation on this study will be over once you have completed the planned study visits at 12 months after the last T-cell infusion.

Long-Term Follow-Up:

For safety reasons, the U.S. Food and Drug Administration (FDA) requires that patients who receive infusions of stem cells treated with a gene transfer procedure must have long-term follow-up for at least 15 years after receiving the gene transfer. You will be asked to sign a separate consent form for a long-term follow-up study named Protocol 2006-0676.

This is an investigational study. The gene transfer (or T-cell infusion with genetically changed T cells) is not commercially available or FDA approved for use in this type of disease. Gene transfer in this study is considered investigational.

Up to 54 patients will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Patients with a history of CD19+ lymphoid malignancies that are primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease.
2. Patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: 1) Acute Lymphoblastic Leukemia (ALL) with translocations 9;22 or 4;11, hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure,and/or evidence of minimal residual disease; or, 2) Acute biphenotypic leukemia; or, 3) Double hit nonHodgkin's lymphoma; or, 4) Non-Hodgkin's Lymphoma (NHL) in second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant). Double hit lymphomas in first remission or more advanced disease; or, 5) Small Lymphocytic Lymphoma (SLL), or Chronic Lymphocytic Leukemia (CLL) with progressive disease following standard therapy.
3. Age 1 to 75 years old.
4. Performance score of at least 80% by Karnofsky or PS < 3 (ECOG) (age >/= 12 years), or Lansky Play-Performance Scale of at least 60% or greater (age <12 years).
5. Two Cord Blood units identified that are matched with the patient at 4/6, 5/6, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 1.5 x 10^7 total nucleated cells/Kg recipient body weight (pre-thaw). One Cord Blood unit may be used (in lieu of two) if it contains at least 2.5 x 10^7 total nucleated cells/Kg recipient body weight (pre-thaw).
6. Have identified a back up cells source in case of engraftment failure. The source can be autologous, related or unrelated.
7. Cardiac Function: left ventricular ejection fraction >/= 40%.
8. Pulmonary function: forced expiratory volume at one second (FEV1), forced vital capacity (FVC) and diffusing capacity of lung for carbon monoxide (DLCO) >/= 50% of expected, corrected for hemoglobin. For children </= 7 years of age who are unable to perform pulmonary function testing (PFT), oxygen saturation >/= 92% on room air by pulse oximetry.
9. Renal function: Serum creatinine \