Donor Stem Cell Transplant in Treating Patients With Myeloid Cancer or Other Disease

Inclusion Criteria: * Primary acute myeloid leukemia (AML) * First complete remission (CR) with high risk features as defined by: failure to achieve remission by day 21 after induction chemotherapy, or the presence of chromosomal abnormalities involving any of the following: -5/de (5q), -7/del(7q), inversion 3q, abnormalities of 11q23, 20q, 21q, del(9q), translocation 6;9, translocation 9;22, abnormalities of 17p, or complex karyotype with \> or = 3 abnormalities. Complete remission is defined as \< 5% blasts in the marrow. * Second CR or subsequent in remission * Refractory or relapsed disease with absolute peripheral blood blasts \< 2000/mcL * Secondary AML in remission or relapse * Chronic myelogenous leukemia (CML) in accelerated or blast phase * Accelerated phase is defined by any one of the following: * Blasts 10% to 19% of peripheral blood white cells or bone marrow cells * Peripheral blood basophils at least 20% * Persistent thrombocytopenia (\<100 x 10\^9/L) unrelated to therapy, or persistent thrombocytosis (\>1000 x 10\^9/L) unresponsive to therapy * Increasing spleen size and increasing white blood cell (WBC) count unresponsive to therapy * Cytogenetic evidence of clonal evolution (i.e., the appearance of an additional genetic abnormality that was not present in the initial specimen at the time of diagnosis of chronic phase CML) * Resistance to tyrosine kinase inhibitors (imatinib or other) defined as no complete cytogenetic response even if the above criteria are not met. * Blast phase is defined by either of the following: * Blasts 20% or more of peripheral blood white cells or bone marrow cells * Extramedullary blast proliferation * Large foci or clusters of blasts in bone marrow biopsy * Primary myelodysplastic syndrome (MDS) with an IPSS score \>1 * Secondary MDS with any international prostate symptom score (IPSS) * Age ≤60 years * Co-Morbidity score 0-2 * At least 35 days following start of preceding leukemia induction therapy Exclusion Criteria: * Patients for whom a suitable HLA genotypically identical sibling or fully matched HLA-A, -B, -C, and -DRB1 unrelated donor is available. * Patients greater than 60 years of age. * Hypersensitivity to thymoglobulin. * Symptomatic uncontrolled coronary artery disease or congestive heart failure. * Hepatic disease with transaminases or bilirubin \> 2 times upper limit of normal (ULN) except for isolated hyperbilirubinemia attributed to Gilbert's syndrome. * Severe hypoxemia with room air - Partial Pressure of Oxygen in Arterial Blood - (PAO2) \< 70, supplemental oxygen-dependence, or carbon monoxide diffusing capacity (DLCO) \< 50% predicted. * Impaired renal function with creatinine \> 2 times upper limit of normal (ULN) or creatinine clearance measured by 24-hour urine collection \< 50% normal for age, gender, and weight. * Patients with central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy. * Patients who are human immunodeficiency virus (HIV) seropositive. * Patients who are pregnant or breast-feeding. * Patients with active infections that are untreated, or failing to respond to appropriate therapy. * Karnofsky performance status \< 50%. * Prior allogeneic or autologous bone marrow, peripheral blood stem cell, or umbilical cord blood transplant. * Inability to provide informed consent. * Co-morbidity score \>2 * Less than 35 days from start of previous leukemia induction therapy