Umbilical Cord Blood (UCB) Transplantation in Pediatric Patients With High Risk Leukemia and Myelodysplasia

Inclusion Criteria: * Patients must be up to 21 years of age * Patients cannot receive total body irradiation (TBI) because of: * Young age - \< 2 years at diagnosis of leukemia resulting in an age \< 4 years at transplantation (due to risk of severe growth retardation and brain damage). * Inability to tolerate TBI because of prior radiation or organ toxicity. * Refractory/multiply relapsed leukemia, for which a traditional TBI/cyclophosphamide regimen would unlikely lead to a successful outcome. * Patients must have a partially human leukocyte antigen (HLA)-matched UCB unit. Unit must be HLA-matched minimally at 4 of 6 HLA-A and B (at intermediate resolution by molecular typing) and DRB1 (at high resolution by molecular typing) loci with the patient. The unit must deliver a pre-cryopreserved nucleated cell dose of at least 2.5 x 10\^7 per kilogram. * Acute myelogenous leukemia (AML) at the following stages: * High risk first complete remission (CR1), defined as: * Having preceding myelodysplasia (MDS) * High risk cytogenetics (High-risk cytogenetics: del (5q) -5, -7, abn (3q), t (6;9) complex karyotype (\>= 5 abnormalities) * Requiring \> 2 cycles chemotherapy to obtain CR; * Second or greater CR. * First relapse with \< 25% blasts in bone marrow. * Patients with therapy-related AML whose prior malignancy has been in remission for at least 12 months. * Acute lymphocytic leukemia (ALL) at the following stages: * High risk first remission, defined as: 1. Ph+ ALL; or, 2. Myeloid/lymphoid leukemia (MLL) rearrangement with slow early response \[defined as having M2 (5-25% blasts) or M3 (\>25% blasts on bone marrow examination on Day 14 of induction therapy)\]; or, 3. Hypodiploidy (\< 44 chromosomes or DNA index \< 0.81); or, 4. End of induction M3 bone marrow; or, 5. End of induction M2 with M2-3 at Day 42. * High risk second remission, defined as: 1. Bone marrow relapse \< 36 months from induction; or, 2. T-lineage relapse at any time; or, 3. Very early isolated central nervous system (CNS) relapse (6 months from diagnosis); or, 4. Slow reinduction (M2-3 at Day 28) after relapse at any time. * Any third or subsequent CR. * Biphenotypic or undifferentiated leukemia in any CR or if in 1st relapse must have \< 25% blasts in bone marrow (BM). * MDS at any stage. * Chronic myelogenous leukemia (CML) in chronic or accelerated phase. * All patients with evidence of CNS leukemia must be treated and be in CNS CR to be eligible for study. * Patients ≥ 16 years old must have a Karnofsky score ≥ 70% and patients \< 16 years old must have a Lansky score ≥ 70%. * Signed informed consent. * Patients with adequate physical function as measured by: 1. Cardiac: Left ventricular ejection fraction at rest must be \> 40%, or shortening fraction \> 26% 2. Hepatic: Bilirubin ≤ 2.5 mg/dL; and alanine transaminase (ALT), aspartate transaminase (AST) and Alkaline Phosphatase ≤ 5 x upper limit of normal (ULN) 3. Renal: Serum creatinine within normal range for age, or if serum creatinine outside normal range for age, then renal function (creatinine clearance or GFR) \> 70 mL/min/1.73 m2. 4. Pulmonary: Diffusing capacity of the lungs for carbon monoxide (DLCO), Forced expiratory volume in 1 second (FEV1), Forced vital capacity (FVC) (diffusion capacity) \> 50% of predicted (corrected for hemoglobin); if unable to perform pulmonary function tests, then O2 saturation \> 92% of room air. Exclusion Criteria: * Pregnant (B-positive HCG) or breastfeeding. * Evidence of HIV infection or HIV positive serology. * Current uncontrolled bacterial, viral or fungal infection (currently taking medication and progression of clinical symptoms). * Autologous transplant \< 6 months prior to enrollment. * Prior autologous transplant for the disease for which the UCB transplant will be performed. * Allogeneic hematopoietic stem cell transplant \< 6 months prior to enrollment. * Active malignancy other than the one for which the UCB transplant is being performed within 12 months of enrollment * Active CNS leukemia. * Requirement of supplemental oxygen. * HLA-matched related donor able to donate.