Low-Dose Fludarabine, Busulfan, and Anti-Thymocyte Globulin Followed By Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematologic Cancer

OBJECTIVES: Primary * Assess the feasibility of performing umbilical cord blood transplants in older patients or younger infirm patients with advanced hematologic malignancies using a reduced-intensity preparative regimen, as determined by \> 80% engraftment rate at day 180 and a \< 50% transplant-related mortality rate at day 100. Secondary * Describe the time to neutrophil and platelet recovery in patients treated with this regimen. * Determine disease-specific, event-free, and overall survival rate at days 180 and 360. * Determine the incidence, severity, and timing of acute and chronic graft-versus-host disease in patients treated with this regimen. * Evaluate T-cell, B-cell, and natural killer cell recovery in patients treated with this regimen. * Assess lineage-specific chimerism after transplantation and describe the contribution of each individual cord blood unit to post-transplantation hematopoiesis. OUTLINE: This is a pilot study. * Reduced-intensity preparative regimen: Patients receive fludarabine IV over 30 minutes on days -8 to -4, busulfan IV over 2 hours 4 times daily on days -4 and -3, and anti-thymocyte globulin IV over 6 hours on days -3 to -1. * Allogeneic umbilical cord blood transplantation: Patients undergo allogeneic umbilical cord blood transplant on day 0. Patients receive sargramostim (GM-CSF) subcutaneously or IV beginning on day 7 and continuing until blood counts recover. * Graft-versus-host disease (GVHD) prophylaxis: Patients receive tacrolimus IV continuously over 24 hours or orally (as tolerated) beginning on day -2 and continuing for approximately 9 months. Patients also receive oral mycophenolate mofetil twice daily on days 1-50. After completion of study treatment, patients are followed periodically for 2 years. PROJECTED ACCRUAL: A total of 10 patients will be accrued for this study.