The purpose of this study is to determine the safety of sequential cord blood transplantation (2 cord blood units) for patients who have diseases that are capable of being cured by allogeneic stem cell transplant but do not have a matched family or volunteer unrelated donor.

Eligible patients will receive conditioning therapy with fludarabine 30 mg/m2/day x 6 days, melphalan 100 mg/m2/day x 1 day, rabbit antithymocyte globulin 1.5 mg/kg/day x 4 days. GVHD prophylaxis will consist of cyclosporine starting on day -1 and mycophenolate mofetil starting on day 0. Sequential cord blood units will be infused on Day 0.

Inclusion Criteria:

• Disease Status: NHL, HD, or MM refractory to chemotherapy or relapsed; CLL Rai Stage III/IV, or lymphocyte doubling time of 6 months, or Stage I/II resistant to 2 chemotherapy regimens; AML or ALL in second or subsequent remission or in first remission with adverse cytogenetics or antecedent hematologic disorder
• Estimated disease-free survival of less than one year
• ECOG performance status of 0, 1, 2
• Lack of 6/6 or 5/6 HLA matched related or 6/6 matched unrelated donor, or a donor is not available within the time frame necessary to perform a potentially curative stem cell transplant
• 4/6 or greater HLA A, B, CRB1 cord blood units with a combined nucleated cell dose from of > 3.7 x 10(7) NC/kg

Exclusion Criteria:

• Cardiac disease: symptomatic congestive hearth failure or RVG or echocardiogram determined left ventricular ejection fraction of < 45%, active angina pectoris, or uncontrolled hypertension
• Pulmonary Disease: severe chronic obstructive lung disease, or symptomatic restrictive lung disease, or corrected DLCO of < 50%
• Renal disease: serum creatinine > 2.0 mg/dl
• Hepatic disease: serum bilirubin > 2.0 mg/dl (except in the case of Gilbert's syndrome), SGPT or SGOT > 3 x normal
• Neurologic disease: symptomatic leukoencephalopathy, active DNS malignancy or other neuropsychiatric abnormalities believed to preclude transplantation
• HIV antibody or Hepatitis B surface antigen positivity
• Uncontrolled infection
• Pregnancy or breast-feeding mother