The purpose of this study is to measure the effectiveness of 2 drugs, tacrolimus and sirolimus, in preventing graft versus host disease (GVHD) after treatment with chemotherapy followed by donor cord blood transplantation.

• The chemotherapy portion of the study involves the intravenous administration of fludarabine, for six days (Days 8, 7, 6, 5,4, and 3) before transplant, melphalan, for one day (Day 2) before transplant. Antithymocyte globulin, or thymoglobulin, will be given IV daily for 4 days (days 7, 5, 3, and 1 before transplant). This drug also helps to suppress the immune system, allowing the cord blood cells to grow and reproduce.
• Immunosuppression therapy consists of the drugs tacrolimus and sirolimus. The patient will receive these 3 days before the transplant and every day for 3-6 months after transplant. After the first 100 days post transplant, the doses of tacrolimus and sirolimus will begin to be reduced with the goal of having the patient off both drugs by 6-9 months after transplant.
• After completion of conditioning therapy described above, the patient will receive 2 cord blood units 1-6 hours apart. To help with engraftment, the patient will also receive G-CSF starting on day five after transplant, until the patients white blood cells recover.
• Follow-up visits will continue every 6 months after the last treatment dose and will last up to 2 years.
• Blood tests will be drawn frequently to test whether the donor's immune cells have engrafted as well as to test the levels of Tacrolimus and Sirolimus.

Inclusion Criteria:

• Patients with hematologic malignancies for whom allogeneic stem cell transplantation is deemed clinically appropriate
• Non-Hodgkin's lymphoma, or Hodgkin's lymphoma: in Complete Remission >2 (second complete remission, third complete remission, etc) or in partial remission
• Multiple myeloma: relapsed
• Chronic lymphocytic leukemia, Rai stage III or IV, or lymphocyte doubling time of 6 months, or stage I-II, having progressed after > 2 chemotherapy regimens, in partial remission.
• Acute myelogenous or lymphoblastic leukemia in second or subsequent remission or in first remission with adverse cytogenetic or antecedent hematologic disorder
• Chronic myelogenous leukemia in accelerated or second stable phase, or imatinib resistant and not eligible for an ablative transplant
• Myelodysplasia, previously treated or not eligible for ablative transplant
• Age 18-65 years.
• ECOG performance status of 0, 1, or 2.
• Lack of 6/6 or 5/6 HLA-matched related, 10/10 matched unrelated donor, or unrelated donor not available within the time frame necessary to perform a potentially curative stem cell transplant.

Exclusion Criteria:

• Cardiac disease:

  • symptomatic congestive heart failure or
  • radionuclide ventriculogram (RVG) or echocardiogram determined left ventricular ejection fraction of < 40%,
  • active angina pectoris, or
  • uncontrolled hypertension.

• Pulmonary disease:

  • severe chronic obstructive lung disease, or
  • symptomatic restrictive lung disease, or
  • corrected DLCO of < 50% of predicted.

• Renal disease:

  • serum creatinine > 2.0 mg/dl.

• Hepatic disease:

  • serum bilirubin > 2.0 mg/dl (except in the case of Gilbert's syndrome or hemolytic anemia in which the bilirubin can be elevated greater than 2.0mg/dl),
  • SGOT or SGPT > 3 x normal.

• Neurologic disease:

  • symptomatic leukoencephalopathy,
  • active central nervous system (CNS) malignancy or other neuropsychiatric abnormalities believed to preclude transplantation (previous CNS malignancy, presently in complete remission [CR] is not exclusion).

• HIV antibody.

• Uncontrolled infection.

• Pregnancy or breast feeding mother.