Infusion of Expanded Cord Blood T Cells

Inclusion Criteria:

1. Patient must have one of the following hematologic malignancies:

   * AML
   * ALL
   * CML
   * NHL
   * HD, or
   * CLL (Select from Criteria #2 through #7)
2. Acute Myelogenous Leukemia (AML), Myelodysplastic Syndrome (MDS; Myelodysplastic syndromes with International Prognostic Scoring System score \>2 or myelodysplasia that has not responded to chemotherapy): induction failure, high-risk for relapse 1st remission (with high-risk cytogenetics or FLT3 mutation), 2nd or 3rd complete remission, or 2nd relapse with less than 10% blasts in the bone marrow and/or peripheral blood.
3. Acute Lymphoblastic Leukemia (ALL): induction failure, 1st complete remission with Philadelphia chromosome or translocation, 2nd or 3rd complete remission, or 2nd relapse with less than 10% blasts in the bone marrow and/or peripheral blood.
4. Chronic Myelogenous Leukemia (CML) second chronic phase, accelerated phase or blast crisis with less than 10% blasts in the bone marrow and/or peripheral blood and failure of at least one tyrosine kinase inhibitor.
5. Non-Hodgkin's Lymphoma (NHL): Induction failures, second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant).
6. Hodgkin's Disease (HD): Induction failures, second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant).
7. Chronic Lymphocytic Leukemia (CLL): Failure of one multi-agent regimen including fludarabine or other nucleoside analog
8. Patient's Age Criteria: Age \>/= 6 months and \</= 55 years
9. Performance score of \< 3 (Zubrod score).
10. Adequate major organ system function as demonstrated by: (SEE CRITERIA #11 through #14)
11. Left ventricular ejection function of \>/= 50%.
12. Pulmonary function test demonstrating a diffusion capacity of at least 50% predicted. If unable to perform pulmonary function test (most children \<6 years of age), pulse oximetry \>/= 92% on room air.
13. Creatinine \</= 1.6 mg/dL in adults and \</=2 times upper limit of normal in pediatric patients.
14. SGPT/bilirubin \</= 3.0 x normal.
15. Signed informed consent.
16. Negative Beta HCG or urine test in females of childbearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on the study.
17. Cord Blood Requirements: Unrelated CB will be used as a source of hematopoietic support if a 5/6 or 6/6 related or 6/6 unrelated bone marrow donor is not available, or if the tempo of the patient's disease dictates it is not in the patient's best interest to wait for an unrelated marrow donor to be procured. The back up cord blood unit must match at 4 of 6 HLA.
18. An unrelated donor, second cord blood transplant, or autologous stem cells harvested prior to high-dose chemotherapy will be used in the event of graft failure. Based on the health status of the patient, autologous harvest may or may not occur. To be determined by the treating physician. The cord blood unit will be identified prior to enrollment in this study.

Exclusion Criteria:

1. HIV positive (due to the extreme immunosuppressive nature of allogeneic stem cell transplant)
2. Patient with active (untreated) CNS disease
3. Availability of an appropriate, willing, HLA-matched related marrow donor
4. Active invasive infections.
5. Pregnant or breast-feeding