Treatment of Hematologic Malignancies With Single-Unit or Double-Unit Cord Blood Transplantation

Inclusion Criteria: Patients must: * Be between the ages of 18 and 49 years inclusive. * Have histologically proven hematologic malignancy and must meet accepted indications for allogeneic stem cell transplantation: * Acute myeloid leukemia (AML): high-risk in first complete remission (CR), near first CR (\< 20% marrow blasts after induction therapy), or beyond first remission. * Acute lymphoblastic leukemia with high-risk features in first remission (Philadelphia chromosome positive or other similar high-risk features), near first CR (\< 20% marrow blasts after induction therapy), or beyond first remission. * Chronic myeloid leukemia: in chronic phase (CP) and have failed Gleevec or are intolerant and have signs of failing other treatments, or have history of accelerated or blast phases. Storage of autologous peripheral blood stem cells is recommended for this group for patients, particularly if in chronic phase. * Non-Hodgkin's lymphoma: aggressive histology with relapsed or primary refractory disease and not eligible for autologous transplantation. Patients with low-grade histology must have either failed 2 or more lines of systemic chemotherapy including one rituximab-based regimen. Radioimmunotherapy will be considered the equivalent of one line of chemotherapy. Excluded if prior radiotherapy to chest or prior autologous transplantation. * Myelodysplastic syndromes with International Prognostic Scoring System (IPSS) score of 1.5 or greater. * Other hematological malignancies if approved by the pipeline meeting on a case by case basis. Must also be approved by principal investigator. * Have no HLA-matched or 1-antigen mismatched related donors and no HLA-matched unrelated stem cell donors, or delays involved in performing a search are likely to be detrimental to the patient. * Have an ECOG performance status of 0 or 1 at the time of transplant. * Have major end organs (heart, lungs, liver, and kidneys) assessed and deemed adequate to withstand the effects of high-dose therapy planned for this protocol. * Have given voluntary informed consent. Exclusion Criteria: Patients are ineligible for this protocol if they: * Have a co-morbid medical condition, a psychiatric condition, or organ dysfunction that makes them at high-risk for treatment failure, for failed medical compliance, or for regimen-related toxicity from high-dose therapy. * Patients with any of the following will be excluded: * Pulmonary: hemoglobin (Hb)-adjusted diffusing capacity of lung for carbon monoxide (DLCO) \< 60%, forced expiratory volume in 1 second (FEV1) \< 70% of predicted, or receive continuous supplemental oxygen; * Cardiac: left ventricular ejection fraction (LVEF) \< 50% or on any treatment for congestive heart failure; * Renal: serum creatinine greater than 2.0 or calculated creatinine clearance \< 50 cc/min; * Liver: ALT, AST, or serum bilirubin \> 1.5 x upper limit of normal (ULN). Any patient with elevated transaminases should have careful evaluation for the cause and liver biopsy may be required by the principle investigator. An elevated alkaline phosphatase should be fractionated and, if of liver origin, should be evaluated as for transaminases. * Are female and are pregnant, lactating, or have a positive pregnancy test. * Have a history of previous malignancy except for non-melanoma skin cancer and in-situ carcinoma of the cervix, unless the patient has been progression-free for \> 5 years * Are HIV positive * Refractory malignancy: acute leukemia with greater than 30% blasts in bone marrow unless with untreated first relapse of AML or untreated myelodysplastic syndrome evolved to AML. * Acute leukemia with greater than 1000 blasts/ul in peripheral blood. * Uncontrolled central nervous system (CNS) leukemia or lymphoma. * Prior autologous or allogeneic transplantation using a myeloablative regimen. * Uncontrolled hypertension (systolic blood pressure \[SBP\] \> 140, diastolic blood pressure \[DBP\] \> 90) or hypertension requiring \> 2 drugs for good control (SBP \< 130, DBP \< 90). * Invasive mold infection that is uncontrolled or has received less than one month of antifungal therapy.