Patients with cancers of the blood and immune system often benefit from transplants of stem cells from a genetically well-matched sibling. However, severe problems may follow these transplants because of the high-dose chemotherapy and radiation that accompany the procedure. Also, donated immune cells sometimes attack healthy tissues in a reaction called graft-versus-host disease (GVHD), damaging organs such as the liver, intestines and skin. To reduce toxicity of high-dose preparative chemotherapy, this study performs allogeneic transplant after low doses of chemotherapy. In an attempt to improve anti-tumor effects without increasing GVHD, this study uses donor immune cells (Th2 cells) grown in the laboratory; some patients will receive standard donor immune cells (not grown in laboratory). All patients will receive immune modulating drugs sirolimus and cyclosporine to prevent GVHD.