Umbilical Cord Transplantation for the Elderly Population

Inclusion Criteria: * Ages 55-73 * Patients will have one of the following malignancies: * Acute myelogenous leukemia (AML) deNovo in first CR with adverse cytogenetic abnormalities, M0, M6, M7 subtypes, extramedullary disease in remission or high CD34+ disease (\> 50%) * AML in early relapse (5-10% blasts on bone marrow aspirate or biopsy), or beyond CR-1 with no circulating blasts * AML at any time if resulting from a previous myelodysplasia * Acute lymphocytic leukemia or lymphoblastic lymphoma (ALL) in first CR with adverse prognostic features: t (9; 22), extra medullary disease, or mature B cell phenotype * Acute lymphoid leukemia or lymphoblastic lymphoma in early relapse (5- 10% blasts on aspirate), or beyond CR-1 * Acute Undifferentiated Leukemia or biphenotypic leukemia in CR1 or CR2 * Transfusion dependent myelodysplastic syndrome (MDS) or refractory anemia with excess blasts (RAEB) or RAEB-in transition, CMMOL, or any myelodysplasia with 7q-, 5q-, 7-, 5- or resulting from prior anti cancer therapy. * Relapsed Non-Hodgkin's Lymphoma (NHL), including those that have relapsed after an autologous marrow/blood stem cell transplant * Chronic lymphocytic leukemia (CLL) patient who has had fludarabine and either failed or relapsed. Prior autologous transplant patients are eligible. * Patients with adequate organ function and performance status criteria * Subject must have at least one or the following back-up stem cell sources in case of engraftment failure: * Subject is willing to undergo BM harvest or peripheral blood progenitor cells (PBPC) collection for use in case of engraftment failure (when clinically applicable). * Subject has a second CBU as a possible back up. * Subject's haploidentical family member has been identified and agreed (by signing a written informed consent) to donate hematopoietic stem cells in case of engraftment failure. * Evaluation by social service/psychologist * Subject signs the written informed consent after being aware of the nature of the subject's disease and willingly consents to the treatment program after being informed of alternative treatments, potential risks, benefits and discomforts. * Ability to understand and agree to compliance with strict evaluation, isolation,and medication schedules * Designated primary care giver. * Dental evaluation/treatment completed. * ENT evaluation/treatment completed. * All patient who survive to day 90 are eligible for measurement of T and B cell function and lymphocyte subset numbers to determine immune reconstitution post UCB transplantation with or without StemEx® Exclusion Criteria: * Patient with suitable related donor as defined per institutional guidelines * Chemotherapy resistant or active AML, ALL, AUL, biphenotypic leukemia * AML evolved from myelofibrosis * MDS with 20% or greater bone marrow blasts at pre-transplant workup. Patients may receive therapy and if in remission, are eligible * Prior allogeneic hematopoeitic stem cell transplant at any time * Less than twenty-one days have elapsed since the subject's last radiation or chemotherapy prior to conditioning (except for hydroxyurea) * Uncontrolled bacterial, fungal or viral infection at the time of study enrollment * Seropositive or NAT positive for HIV, HTLV-1 and Hepatitis C * Subjects with signs and symptoms of active central nervous system (CNS) disease * Females who are pregnant or breastfeeding * Allergy to bovine proteins or to aminoglycoside antibiotics (e.g. gentamicin) or to any product, which may interfere with the treatment. * Patient unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, follow-up and research tests. * Enrolled in another clinical trial or received an investigational treatment during the last 30 days, unless approved by the primary investigator.