Basiliximab #2: In-Vivo Activated T-Cell Depletion to Prevent Graft-Versus_Host Disease (GVHD) After Nonmyeloablative Allotransplantation for the Treatment of Blood Cancer

Inclusion Criteria: * Acute myelogenous leukemia: * Second or subsequent remission; patient over 18 yrs of age. * Relapsed after autologous HC transplant, over 18 years of age. * First remission, Philadelphia chromosome + over age 18. * Secondary AML, in first or subsequent remissions. * Acute lymphocytic leukemia: * Philadelphia chromosome + over the age of 50, first or subsequent remission. * Relapse following Autologous HC transplantation, ages over 50. * Second or subsequent remission over the age of 50 * Chronic myelogenous leukemia: * First or second chronic phase over the age of 18. * Accelerated phase over the age of 18. * Must have failed or been intolerant to a standard tyrosine kinase inhibitor. * Chronic lymphocytic leukemia: * Failed nucleoside-based therapy, ages \>18. * Myelodysplasia: * All-risk categories, age greater than 18. * Non-Hodgkin's Lymphoma, less than 76 years of age * Relapsed diffuse aggressive NHL (intermediate and high grade) that fails to achieve CR or PR to conventional salvage chemotherapy. * Aggressive NHL includes diffuse large B cell lymphoma, diffuse mixed small and large cell lymphoma, follicular lymphoma for grade 3 (follicular large cell lymphoma), T or B cell lymphoblastic lymphoma, diffuse small noncleaved (Burkett's or Burkett-like ) lymphoma, mantle cell lymphoma, peripheral T cell lymphoma, anaplastic large cell lymphoma, and other diffuse aggressive lymphomas that are not otherwise classifiable * Aggressive NHL that has relapsed following autologous HCT. Patients that respond to additional treatment for post-transplant relapse are eligible. * Aggressive NHL that does not achieve CR or PR with primary chemotherapy (i.e., primary induction failure). * Low-grade lymphoma refractory to standard therapy, including the following: 1. small cell lymphocytic lymphoma, 2. follicular lymphoma of grades 1 and 2 (follicular small cleaved and follicular mixed small and large cell lymphoma) 3. marginal cell lymphoma, splenic lymphoma), 4. lymphoplasmacytic lymphoma and 5. other lymphomas not otherwise classifiable. * Patients with low-grade lymphoma must have experienced progressive disease after receiving three or more of the following regimens: * alkylator-based therapy (cyclophosphamide/ vincristine/ prednisone) chlorambucil, monoclonal antibody based therapy (e.g., rituximab, Campath-1H, radiolabelled CD20+ antibodies); * nucleoside analog-based therapy (e.g., fludarabine, cladribine).) * Patients with marginal zone lymphoma or gastric MALT type associated with Helicobacter pylori infection must have progressed after receiving appropriate antibiotic therapy as well as three or more regimens as described above * Mantle cell, ages 18-75. * Hodgkin's Disease, ages 18-75. * Relapsed or refractory disease after autologous transplant. * Multiple Myeloma, ages 18-75 * Recurrent disease after two medical therapies * Relapse following autologous transplant * Myelofibrosis, age greater than 18 years * Severe aplastic anemia (refractory to immunosuppressive therapy); age greater than 18 years * Patients with aplastic anemia must have marrow cellularity ≤ 10% plus 2 of the following: 1. Absolute granulocyte count \<500/mm3 2. Corrected reticulocyte count \<1% 3. Untransfused platelet count \<20,000/mm3 on at least 2 occasions 4. Hemoglobin \<9 g/dL (adults) or \< 8 g/dL (children) on at least 2 occasions * Paroxysmal nocturnal hemoglobinuria; age greater than 18 years. * Renal function: creatinine greater than 2.5 * Donor Requirement: * Must have a fully HLA-matched (10 of 10 Antigen matched) related or unrelated donor, eighteen years of age or older, who is capable of undergoing GCSF mobilization and apheresis Exclusion Criteria: * Active CNS disease (the presence of leukemic blasts in the CSF) * Pregnancy or breast-feeding. * Inability to give informed consent. * AST, ALT, total bilirubin \>3x upper limit of normal. * Creatinine \> 2 or creatinine clearance \< 50mL/hr. If patient has a creatinine of \> 2 or creatinine clearance \< 50mL/hr and it is due to the disease process then the patient will not be excluded based on this. * Fractional shortening by echocardiogram not within normal limits per institution or LVEF of \< 40 %. * Pulmonary function: DLCO not within institutional normal limits or DLCO less than 45% of normal predicted, corrected for anemia * Prior allogeneic transplant.