History of allogeneic organ transplantation CRITERIA FOR ALLOGENEIC TRANSPLANTATION Prior autologous or allogeneic organ or tissue transplantation History of allogeneic organ transplantation. Patients who are candidates for allogeneic transplantation, have a suitable donor, and are willing to undergo transplantation Anticipated liver transplantation Prior allogeneic transplantation Previous allogeneic or autologous hematopoietic cell transplantation or solid organ transplantation Patients with prior orthotropic liver transplantation History of allogeneic organ transplantation Allogeneic hematopoietic cell transplantation within 5 years of study drug administration Patients who have undergone prior liver transplantation are ineligible PHASE I: Patients who are between T+40 and T+100 after allogeneic transplantation PHASE I: Patients who have evidence of donor chimerism after allogeneic transplantation PHASE II: Patients who are between T+40 and T+100 after allogeneic transplantation PHASE II: Patients who have >= 80% donor chimerism after allogeneic transplantation ADDITIONAL INCLUSION CRITERIA PERTINENT ONLY FOR PATIENTS WITH PRIOR ALLOGENEIC TRANSPLANTATION: Prior hematopoietic transplantation is allowed (autologous and/or allogeneic) Patients with any of the following hematologic malignancies who are considered to be eligible for allogeneic transplantation: Prior history of allogeneic hematopoietic cell transplantation Patients who have undergone prior liver transplantation are ineligible Relapsed after allogeneic transplantation Prior hematopoietic transplantation is allowed (autologous and/or allogeneic) Use of any of the following after transplantation and prior to starting study therapy: Prior high dose chemotherapy for autologous hematopoietic cell transplantation or prior allogeneic transplantation Relapsed malignancy after transplantation Prior allogeneic or autologous transplantation Prior allogeneic hematopoietic progenitor cell transplantation Transplantation for AML (allogeneic or autologous) allowed unless within 90 days of study entry Relapsed after allogeneic transplantation Patients with B-cell hematological malignancies who are eligible for allogeneic transplantation At least 60 days from day of transplantation Prior autologous or allogeneic organ or tissue transplantation Patients considered for transplantation must have a sufficient graft as based on current criteria of the University of Minnesota Blood and Marrow Transplantation Program Time available (at least 12 weeks) for treatment of hepatitis C prior to autologous or allogeneic transplantation History of allogeneic organ transplantation History of allogeneic organ transplantation. Prior autologous or allogeneic organ or tissue transplantation Prior autologous or allogeneic transplantation for any disease No prior autologous or allogeneic organ or tissue transplantation History of allogeneic organ transplantation. Prior hematopoietic cell transplantation allogeneic or autologous (A prior autologous HCT will be allowed as long as it was part of tandem transplantation). Prior allogeneic hematopoietic cell transplantation ALLOGENEIC TRANSPLANTATION: Subjects who are candidates for allogeneic transplantation, have a suitable donor, and are willing to undergo transplantation Previous liver transplantation Prior allogeneic transplantation Patients who have undergone liver transplantation are excluded Prior allogeneic transplantation Patients must be between 2-6 months post-transplantation at the time of study registration Patients who have received liver transplantation History of allogeneic organ transplantation; History of allogeneic organ transplantation Subjects who relapse after allogeneic transplantation; Patients who have undergone prior liver transplantation are ineligible Prior liver transplantation and on immunosuppression Patients that underwent allogeneic transplantation as a treatment of graft failure Hepatitis A, B and C status will be tested prior to therapy, but results will not exclude patients from participation (if positive, patients will be told they are at higher risk of adverse effects from allogeneic transplantation) No prior autologous or allogeneic hematopoietic cell transplantation Patients that underwent allogeneic transplantation as a treatment of graft failure Patient must meet eligibility criteria for allogeneic transplantation Prior allograft transplantation including liver transplantation. Immunocompromised, as defined by one of the following: Autologous or Allogeneic hematopoietic cell transplantation (HSCT); Lung or lung-heart transplantation; Subjects treated with chemotherapy for hematologic malignancies; Subjects treated with chemotherapy for solid tumor malignancies Has received a prior allogeneic organ or tissue transplantation The participant has received a prior autologous or allogeneic organ or tissue transplantation. Patients who have undergone prior allogeneic transplantation are eligible provided that their transplant day 0 is > 6 months from their first dose of study drug Transplantation of PBSC Patients must have undergone an autologous transplant =< 12 months prior to allogeneic transplantation or have received multi-agent or immunosuppressive chemotherapy within 3 months of the preparative regimen Prior allogeneic transplantation Patients must have a hematologic malignancy for which allogeneic hematopoietic peripheral blood cell transplantation is deemed clinically appropriate. Eligible diseases and stages include the following: Prior allogeneic hematopoietic cell transplantation (HCT) Hematologic disorder requiring allogeneic hematopoietic cell transplantation Transplantation with PBSC Prior allogeneic transplantation