Inclusion Criteria:

- Patients must have one of the following disease types:

- Acute myeloid leukemia (AML) with disease features as described in the protocol.

- Acute lymphoblastic leukemia (ALL) with disease features as described in the
protocol.

- Myelodysplasia with disease features as described in the protocol.

- Chronic myelogenous leukemia (CML) with disease features as described in the
protocol.

- Patients with aggressive non-Hodgkin's lymphoma (NHL), including diffuse large
cell lymphoma, mediastinal B-cell lymphoma, transformed lymphoma, mantle cell
lymphoma, and peripheral T cell lymphoma, who also have one of the disease
features as described in the protocol.

- At least 35 days following start of preceding leukemia induction cytotoxic
chemotherapy.

- For patients in remission, there should be no readily available consenting
HLA-matched related donor who is either matched fully matched or mismatched at only
one locus of HLA-A, -B, and DRB1.

- No availability of a readily available HLA-matched volunteer unrelated donor (8 of 8
allele match at HLA-A, -B, -C and -DRB1).

- Patients must have a matched or partially matched UCB unit with >/= 2.5 x10^7
nucleated cells/kg of recipient weight at the time of cryopreservation.

- No current uncontrolled bacterial, viral or fungal infection (defined as currently
taking medication and progression of clinical symptoms).

- No HIV disease.

- Non pregnant and non-nursing.

- Required baseline laboratory values as described in the protocol.

- Signed written informed consent.

Exclusion Criteria:

- Symptomatic uncontrolled coronary artery disease or congestive heart failure.

- Severe hypoxemia with room air PaO2<70, supplemental oxygen dependence, or DLCO<50%
predicted.

- Patients with central nervous system (CNS) involvement refractory to intrathecal
chemotherapy.

- Prior allogeneic or autologous hematopoietic stem cell transplant in the last 6
months.

- Patients who are taking other insulin secretagogues and/or insulin.

- Patients who have hypersensitivity to sitagliptin.

- Patients with a history of pancreatitis, cholelithiasis, alcoholism, or fasting
hypertriglyceridemia (> 2 x ULN).