Conditioning Treatment With Umbilical Cord Blood Transplant for Hematologic Malignancies

Inclusion Criteria: Subjects 18-70 years old. ECOG 0-2 Patients must have a diagnosis of one of the following: * Chronic Myeloid Leukemia * Acute Myeloid Leukemia * Acute Lymphoblastic Leukemia * Hodgkin's Disease * Non-Hodgkins Lymphoma * Myelodysplastic Syndromes * Myeloproliferative Disorder Patients must have adequate visceral organ function * Patients must furnish written informed consent and HIPAA authorization for release of personal health information. * Patients must be able to understand the requirements of the study, abide by the study restrictions, and agree to return for the required assessments. Exclusion Criteria: * Patients who have a histocompatible sibling-matched donor age 18 to 65 years in good health who is willing to donate stem cells are ineligible. * Patients who are pregnant are ineligible. * Patients are ineligible if they have received cumulative chemotherapy doses in excess of: carmustine (BCNU) 400 mg/m2, and/or a cumulative anthracycline exposure in excess of 550 mg/m2 doxorubicin (Adriamycin®) unless gated-pool radionuclide cardiac scan shows greater than/equal to 45% ejection fraction. Patients who are HIV or HTLV-I, -II antibody sero-positive are ineligible.