Inclusion Criteria:

- Two partially HLA-matched UCB units. Units 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, and the units must be
HLA-matched at 3 of 6 HLA- A, B, DRB1 loci with each other (using same resolution of
molecular typing as indicated above). Two appropriately HLA-matched units must be
available such that one unit delivers a pre-cryopreserved nucleated cell dose of at
least 2.5 x 10^7 per kilogram and the second unit at least 1.5 x 10^7 per kilogram.

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

1. High risk first complete remission (CR1), defined as the following:

- Having preceding myelodysplasia (MDS)

- High risk cytogenetics (high risk cytogenetics: del (5q) -5, -7, abn (3q),
t (6;9) complex karyotype [at least 5 abnormalities],)the presence of a
high FLT3 ITD-AR (> 0.4)

- Requiring more than 1 cycle of chemotherapy to obtain complete remission
(CR);

- FAB M6

2. Second or greater CR

3. First relapse with less than 25% blasts in bone marrow

4. Morphologic complete remission with incomplete blood count recovery

- Therapy-related AML for which prior malignancy has been in remission for at least 12
months

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

1. High risk first remission, defined as one of the following conditions:

- Philadelphia chromosome-positive adult lymphoblastic leukemia (Ph+ ALL)

- Mixed lineage leukemia (MLL) rearrangement with slow early response
(defined as having M2 [5-25% blasts] or M3 [more than 25% blasts on bone
marrow examination on Day 14 of induction therapy])

- Hypodiploidy (less than 44 chromosomes or DNA index less than 0.81)

- End of induction M3 bone marrow

- End of induction M2 with M2-3 at Day 42

- Evidence of minimal residual disease (MRD). If a patient's only high risk
criterion is MRD, approval by a protocol chair or protocol officer is
required for enrollment. For COG centers, this will only be for MRD greater
than 1 percent by flow MRD at the end of extended induction.

2. High risk second remission, defined as one of the following conditions:

- Philadelphia chromosome-positive adult lymphoblastic leukemia (Ph+ ALL)

- Bone marrow relapse less than 36 months from induction

- T-lineage relapse at any time

- Very early isolated central nervous system (CNS) relapse (6 months from
diagnosis)

- Slow reinduction (M2-3 at Day 28) after relapse at any time

- Evidence of minimal residual disease (MRD). If a patient's only high risk
criterion is MRD, approval by a protocol chair or protocol officer is
required for enrollment. For COG centers, this will only be for MRD greater
than 1 percent by flow MRD at the end of extended induction.

3. Any third or subsequent CR

- NK cell lymphoblastic leukemia in any CR

- Biphenotypic or undifferentiated leukemia in any CR or if in first relapse must have
less than 25% blasts in bone marrow (BM)

- Myelodysplastic syndrome (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 or older must have a Karnofsky score of at least 70% and
patients younger than 16 years old must have a Lansky score of at least 70%.

- Patients with adequate physical function as measured by:

1. Cardiac: Left ventricular ejection fraction greater than 40% or shortening
fraction greater than 26%

2. Hepatic: Bilirubin no more than 2.5 mg/dL; ALT, AST and ALP no more than 5 times
the upper limit of normal (ULN)

3. Renal: Serum creatinine within normal range for age, or if serum creatinine is
outside normal range for age, then renal function (creatinine clearance or GFR)
greater than 70 mL/min/1.73 m^2

4. Pulmonary: DLCO, FEV1, FEC (diffusion capacity) greater than 50% of predicted
value (corrected for hemoglobin); if unable to perform pulmonary function tests,
then O2 saturation greater than 92% of room air

Exclusion Criteria:

- Pregnant (β-positive human chorionic gonadotropin [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 less than 12 months prior to enrollment

- Prior autologous transplant for the disease for which the UCB transplant will be
performed

- Prior allogeneic hematopoietic stem cell transplant

- Active malignancy other than the one for which the UCB transplant is being performed
within 12 months of enrollment

- Inability to receive TBI

- Requirement of supplemental oxygen

- HLA-matched related donor able to donate