Multi-center, Open-label Randomized Study of Single or Double Myeloablative Cord Blood Transplantation With or Without Infusion of Off-the-shelf ex Vivo Expanded Cryopreserved Cord Blood Progenitor Cells in Patients With Hematologic Malignancies

Inclusion Criteria:

- Acute myeloid leukemia:

- High risk first complete remission (CR1) as evidenced by preceding
myelodysplastic syndromes (MDS), high risk cytogenetics (for example, monosomy 5
or 7, or as defined by referring institution treatment protocol), >= 2 cycles to
obtain complete remission (CR), erythroblastic or megakaryocytic leukemia; >=
second complete remission (CR2)

- All patients must be in CR as defined by hematologic recovery and < 5% blasts by
morphology within the bone marrow and a cellularity of >= 15% for age

- Patients in which adequate marrow/biopsy specimens cannot be obtained to
determine remission status by morphologic assessment, but have fulfilled
criteria of remission by flow cytometry, recovery of peripheral blood counts
with no circulating blasts, and/or normal cytogenetics (if applicable) may still
be eligible; reasonable attempts must be made to obtain an adequate specimen for
morphologic assessment, including possible repeat procedures; these patients
must be discussed with the principal investigator prior to enrollment

- Acute Lymphoblastic Leukemia

- High risk CR1 [for example, but not limited to: t(9;22), t(1;19), t(4;11) or
other mixed-lineage leukemia (MLL) rearrangements, hypodiploid]; greater than 1
cycle to obtain CR; CR2 or greater

- All patients must be in CR as defined by hematologic recovery and < 5% blasts by
morphology within the bone marrow and a cellularity of >= 15% for age

- Patients in which adequate marrow/biopsy specimens cannot be obtained to
determine remission status by morphologic assessment, but have fulfilled
criteria of remission by flow cytometry, recovery of peripheral blood counts
with no circulating blasts, and/or normal cytogenetics (if applicable) may still
be eligible; reasonable attempts must be made to obtain an adequate specimen for
morphologic assessment, including possible repeat procedures; these patients
must be discussed with the principal investigator prior to enrollment

- Chronic myelogenous leukemia excluding refractory blast crisis; to be eligible in
first chronic phase (CP1) patient must have failed or be intolerant to tyrosine
kinase inhibitor therapy

- Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate
(Int)-2 or High risk (i.e., refractory anemia with excess blasts [RAEB], refractory
anemia with excess blasts in transformation [RAEBt]) or refractory anemia with severe
pancytopenia or high risk cytogenetics; blasts must be < 10% by a representative bone
marrow aspirate morphology

- Karnofsky (>= 16 years old) >= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1

- Lansky (< 16 years old) >= 60

- Adults: calculated creatinine clearance must be > 60 mL and serum creatinine =< 2
mg/dL

- Children (< 18 years old): calculated creatinine clearance must be > 60 mL/min

- Total serum bilirubin must be < 3mg/dL unless the elevation is thought to be due to
Gilbert's disease or hemolysis

- Transaminases must be < 3 x the upper limit of normal

- Diffusing capacity of the lung for carbon monoxide (DLCO) corrected > 60% normal

- For pediatric patients unable to perform pulmonary function tests, oxygen (O2)
saturation > 92% on room air

- May not be on supplemental oxygen

- Left ventricular ejection fraction > 45%

- OR shortening fraction > 26%

- Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

- Uncontrolled viral or bacterial infection at the time of study enrollment

- Active or recent (prior 6 month) invasive fungal infection without infectious disease
(ID) consult and approval

- History of human immunodeficiency virus (HIV) infection

- Pregnant or breastfeeding

- If =< 18 years old, prior myeloablative transplant within the last 6 months

- If > 18 years old prior myeloablative allotransplant or autologous transplant

- Extensive prior therapy including > 12 months alkylator therapy or > 6 months
alkylator therapy with extensive radiation