Inclusion Criteria:

          -  Must have a partially HLA matched UCB unit with a pre-cryopreserved TNC dose >2.5 x
             107 per kilogram recipient weight. HLA matching is initially based on 4 of 6 HLA-A and
             B (at low or intermediate resolution by molecular typing) and DRB1 (at high resolution
             by molecular typing).

          -  Eligible Diseases

               -  Acute myelogenous leukemia (AML) at the following stages:

                    -  Intermediate to high risk leukemia in first complete remission (CR1) based
                       on institutional criteria.

                    -  Any second or subsequent CR.

                    -  Secondary AML with prior malignancy that has been in remission for at least
                       12 months.

               -  Acute lymphocytic leukemia (ALL) at the following stages:

                    -  High risk first remission.

                         1. Ph+ ALL, or

                         2. MLL rearrangement with slow early response at Day 14, 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 CR based on institutional criteria (eg, for children, bone
                       marrow relapse <36 months from induction or T-lineage bone marrow relapse or
                       very early isolated central nervous system (CNS) relapse <6 months from
                       diagnosis, or slow re-induction (stage M2-3 at day 28 after induction)
                       regardless of length remission.

                    -  Any third or subsequent CR.

               -  Biphenotypic/undifferentiated leukemia in CR

               -  Chronic myelogenous leukemia (CML) excluding refractory blast crisis

               -  Myelodysplasia (MDS) IPSS Int-2 or High risk (i.e. RAEB, RAEBt) or refractory
                  anemia

          -  Other Inclusion Criteria

               -  Karnofsky score >70% (16 years and older) or a Lansky play score >70 (children
                  <16 years) - appendix II

               -  Adequate organ function defined as:

                    -  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.

                    -  Hepatic: Bilirubin ≤2.5 x mg/dL; AST, ALT, alkaline phosphatase <5 x upper
                       limit of normal,

                    -  Pulmonary function: DLCO, FEV1, FEC (diffusion capacity) >50% of predicted
                       (corrected for hemoglobin); if unable to perform pulmonary function tests,
                       then normal O2 saturation on room air.

                    -  Cardiac: Left ventricular ejection fraction at rest must be >45%

               -  Available 'back-up' HSPC graft (e.g, second partially HLA matched UCB unit,
                  haploidentical related donor).

               -  Voluntary written consent signed (adult or parental) before performance of any
                  study-related procedure not part of normal medical care

        Exclusion Criteria:

          -  Pregnant or breast feeding. The agents used in this study may be teratogenic to a
             fetus and there is no information on the excretion of agents into breast milk. Females
             of childbearing potential must have a blood test or urine study within 14 days prior
             to study enrollment to rule out pregnancy.

          -  Evidence of human immunodeficiency virus (HIV) infection or known HIV positive
             serology.

          -  Active bacterial, viral or fungal infection (currently taking medication and
             progression of clinical symptoms).

          -  Prior autologous or allogeneic transplant within past 12 months.

          -  Other active malignancy.

          -  Inability to receive TBI 1320 cGy (e.g., extensive prior therapy including >12 months
             alkylator therapy or >6 months alkylator therapy with extensive radiation. Or prior
             Y-90 ibritumomab (Zevalin) or I-131 tostumomab (Bexxar), as part of their salvage
             therapy.