A Phase II Study of Umbilical Cord Blood Transplantation

Inclusion Criteria: Patients meeting eligibility criteria for unrelated allogeneic stem cell transplantation may be considered for participation on this clinical trial if and only if they meet each of the following criteria: * Patients must have one of the following diagnoses * Relapsed or refractory hematologic malignancy, or * High risk hematologic malignancy in first remission, or * Refractory acquired marrow failure state, or * Inherited disorder of metabolism or marrow failure state without alternative curative therapy. * Patients must not have a 6/6 or 5/6 HLA-matched related donor. * Patients must not have a HLA-A, -B and -DRB1 high resolution matched unrelated donor following registry search, or cannot (in the opinion of the treating physician) wait the median 3 months to receive a MUD unit. * Patients must demonstrate an ability to understand and willingness to sign the informed consent document Patients considered for myeloablative conditioning must satisfy the following additional criteria: * Patients must be up to age 55 (inclusive) * Patients must have serum direct bilirubin ≤ 2.0 mg/dl and transaminases ≤ 2x institution upper limit of normal * Patients must have serum creatinine ≤ 2 mg/dl with creatinine clearance ≥ 60 ml/min (either calculated or measured). * Patients must have MUGA scan or echocardiogram normal for the institution, but not less than 45% left ventricular ejection fraction and no clinical evidence of cardiac dysfunction. * Patients must have an ECOG performance status of 0 or 1 (see Appendix C). * Patients must have adequate pulmonary function when corrected for hemoglobin and alveolar volume as evidenced by a diffusion capacity and FEV1 \> 50% of predicted. Patients considered for reduced-intensity conditioning must satisfy the following additional criteria: * Patients must not be candidates for myeloablative conditioning due to any one of the following: Prior myeloablative stem cell transplantation, Age \> 50, Co morbid illness * In opinion of treating physician, unable to comply with or withstand rigors of myeloablative conditioning * Patients with leukemia must have circulating and bone marrow blast counts \< 5%, all other patients must have chemotherapy responsive disease * Patients must be between the ages of 18 and 70 (inclusive) * Patients must have serum direct bilirubin ≤ 2.0 mg/dL and transaminases ≤ 3x institution upper limit of normal * Patients must have creatinine clearance ≥ 30 ml/min (either calculated or measured). * Patients must have MUGA scan or echocardiogram documenting left ventricular ejection fraction of no less than 35% and no clinical evidence of cardiac dysfunction. Patients must have an ECOG performance status of 0 or 1 (see Appendix C). Patients must have adequate pulmonary function when corrected for hemoglobin and alveolar volume as evidenced by a diffusion capacity and FEV1 ≥ 40% of predicted. Exclusion Criteria: Patients are ineligible for participation on this trial if they meet any of the following criteria: * Patients with a history of myocardial infarction within the preceding 6 months, significant arrhythmia within the preceding 3 months, uncontrolled hypertension or congestive heart failure are ineligible. * Patients with unstable angina are not eligible. * Pregnant or lactating women are ineligible. * Male and female patients who do not agree to practice approved methods of birth control for the duration of the study are ineligible. * Patients with uncontrolled infection are ineligible. * Patients who are HIV positive or have evidence of chronic viral hepatitis are ineligible. * Patients unable to comply with requirements for compliance with therapeutic plan and/or scheduled evaluations