Infusion of Allogeneic CD19-Specific T Cells From Peripheral Blood

The goal of this clinical research study is to learn if researchers can successfully and safely give HSCT patients an infusion of white blood cells (called T-cells) that have been genetically changed. The process of changing the DNA (the genetic material in cells) of these T-cells is called "gene transfer." Researchers want to learn if these genetically-changed T-cells are effective in attacking cancer cells in patients with advanced B-cell lymphoma or leukemia, after they have received standard allogeneic HSCT. Researchers want to find out the highest dose of these special T-cells that can be given safely to leukemia and lymphoma patients. Researchers also want to learn how long the changed T-cells stay in your body, and if adding them to standard transplant can improve how you respond to treatment.