Phase I Study of Sequential Cord Blood Transplants

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