Participants have a central venous line (large plastic tube) placed into a major vein. This tube stays in the body during the entire treatment period for infusing the donated stem cells and T lymphocytes, giving medications, including chemotherapy and other drugs, antibiotics, and blood transfusions, and withdrawing blood samples. Treatment starts with induction chemotherapy, in which patients receive one or two cycles of the anti-cancer drugs fludarabine and cyclophosphamide. (One cycle consists of 4 days on drug therapy followed by a 17-day rest period.) G-CSF, a drug that boosts white cell production, is also given to reduce the risk of infection. Several days before the transplant procedure, patients begin conditioning chemotherapy with higher doses of cyclophosphamide and fludarabine. Three days after the conditioning therapy is completed, the stem cells are infused. To help prevent both rejection of the donor stem cells and GVHD, patients receive cyclosporine (first by vein and later by mouth) for several weeks after the transplant. Infusions of donor lymphocytes begin about 6 weeks after the transplant to boost the immune system and enhance the graft-versus-tumor effect.