Inclusion Criteria:

- Only patients requiring a double umbilical cord blood (UCB) transplant are to be
considered for this study.

UCB Requirements

- Three UCB units are required - one for Treg production and two for UCB transplant.
The unrelated UCB donors must be 4-6/6 HLA-A, B, DRB1 matched with the recipient (HLA
matching using molecular techniques: A and B to antigen level resolution and DR to
allele level resolution). Suitable UCB units will be selected according to the
University Of Minnesota UCB Graft Selection Algorithm.

- Suitable UCB units must be ABO matched.

Disease Criteria:

- Patients aged 18 to 55 years

- Acute Myeloid Leukemia: with morphologically persistent disease in a representative
bone marrow aspirate sample with ≤ 10% blasts after at least 1 cycles of chemotherapy
(if patient refuses or is disqualified from alternative protocols), or in 3rd or
higher complete remission (CR).

- Acute Lymphocytic Leukemia: with morphologically persistent disease in a
representative bone marrow aspirate sample with ≤ 10% blasts after at least 1 cycles
of chemotherapy, or in 3rd or higher CR

- Chronic Myelogenous Leukemia in Blast Crisis: with ≤10% residual blasts in the bone
marrow aspirate after at least 1 cycle of induction chemotherapy in combination with
a tyrosine kinase inhibitor (TKI)

- Refractory Anemia with Excess Blasts: (≤ 10%) in representative bone marrow aspirate
sample of blasts after 1 cycle of induction chemotherapy. If treated with
hypomethylating agents, patients are eligible if blast count is ≤ 10% after 4 cycles
or evidence of stable or progressive disease after at least 2 cycles.

- Chronic Myeloproliferative Disease

- Chronic Lymphocytic Leukemia/Small Lymphocytic Lymphoma, Marginal Zone B-Cell
Lymphoma or Follicular Lymphoma: disease must be refractory after at least two
chemotherapy regimens or is chemotherapy sensitive but has residual nodal disease of
≥ 5 cm

- Lymphoplasmacytic Lymphoma, Mantle-Cell Lymphoma, Prolymphocytic Leukemia: disease
must be refractory after at least two chemotherapy regimens or is chemotherapy
sensitive but has residual nodal disease of ≥ 5 cm

- Large Cell Non-Hodgkin's Lymphoma: disease must be refractory after at least two
chemotherapy regimens or is chemotherapy sensitive but has residual nodal disease of
≥ 5 cm

- Lymphoblastic Lymphoma, Burkitt's Lymphoma, and other High-Grade NHL: disease must be
refractory after at least two chemotherapy regimens or is chemotherapy sensitive but
has residual nodal disease of ≥ 5 cm

- Performance Status, Age, and Organ Function

- Adequate performance status defined as a Karnofsky score ≥ 80%

- Adequate organ function defined as:

- Renal: creatinine < 2.0 mg/dL,

- Hepatic: bilirubin, AST/ALT, ALP < 5 x upper limit of normal,

- Pulmonary function: DLCOcorr > 50% normal,

- Cardiac: left ventricular ejection fraction > 45%

- Voluntary written informed consent signed before performance of any study-related
procedure not part of normal medical care

Exclusion Criteria:

- Available medically suitable HLA-identical related donor

- Active infection at time of transplantation (including active infection with
Aspergillus or other mold within 30 days)

- History of HIV infection

- 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 registration to rule out pregnancy

- Prior myeloablative transplant within the last 6 months

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

- Patients who have received Y-90 ibritumomab (Zevalin) or I-131 tositumomab (Bexxar)
as part of their salvage therapy (not eligible for myeloablative umbilical cord blood
transplant)