Leukemia and lymphoma are types of blood cancers. Chemotherapy is a common treatment option for people with these types of cancers, but if the cancer does not respond well to chemotherapy, or if the cancer returns, people may need to consider other options. A bone marrow transplant, which is a type of stem cell transplant in which healthy bone marrow is donated to a patient by a related or unrelated donor, is commonly used to treat leukemia and lymphoma. Recently, stem cell transplants using umbilical cord blood have become a viable option to treat these types of cancers. Traditionally, umbilical cord blood, which is the blood left over in the placenta after a baby is born, has been disposed of with the placenta. However, over the past few years, doctors have begun to collect and freeze the umbilical cord blood cells so that they may be used in stem cell transplant procedures at a later time.

Typically, people who are undergoing a stem cell transplant receive high doses of chemotherapy before the transplant to prepare their bodies to accept the donor stem cells. In this study, participants will undergo a new type of stem cell transplant called a nonmyeloablative transplant, which is a reduced intensity method of transplantation that does not require high doses of chemotherapy. The purpose of the study is to examine the safety and effectiveness of a nonmyeloablative stem cell transplant that uses umbilical cord blood as a treatment option for people with leukemia or lymphoma.

This study will enroll people with leukemia or lymphoma. Participants will be admitted to the hospital and will first receive a type of chemotherapy called cyclophosphamide, which will be given intravenously on the sixth day before the transplant. In addition, another type of chemotherapy, fludarabine, will be given intravenously each day for 5 days before the transplant. Three days before the transplant, participants will receive cyclosporine and mycophenolate mofetil (MMF), to help prevent the body from rejecting the stem cells and to help decrease the risk of developing a complication called graft-versus-host-disease (GVHD), which is an attack by the donor cells on the body's normal tissues. Some participants may receive tacrolimus instead of cyclosporine. After 6 days, participants will receive a small dose of radiation. The next day, participants will undergo the umbilical cord blood stem cell transplant.

Participants will remain in the hospital for approximately 2 to 3 months total, but possibly longer if there are complications. Beginning on the first day after the transplant, participants will receive daily injections of a growth factor called granulocyte-colony stimulating factor (G-CSF), which is a natural protein that increases the white blood cell count; G-CSF will be continued until a participant's white blood cell count is normal again. Participants will continue to receive MMF for 30 days and cyclosporine or tacrolimus for 180 days after the transplant. While participants are in the hospital, blood samples will be collected regularly to evaluate the response and possible side effects to treatment, including GVHD. If necessary, participants will receive platelet and red blood cell transfusions. At follow-up study visits 6 months and 1 year after the transplant, blood samples will be obtained. Study researchers will keep track of participants' medical condition through phone calls or mailings to participants and their doctors once a year for the rest of the participants' lives.