We believe that the risks of allogeneic transplant can be drastically reduced if the following criteria can be met: (1) consistent engraftment, (2) little or no GVHD with the ability to rapidly withdraw immune suppression, (3) rapid recovery of CD4 counts to levels greater than 200 cells/micro liter. Our prior (ongoing) trial attempts to address how LLME treated T cells given as donor lymphocyte infusion (DLI) can address points 2 and 3 above. The current study addresses how treatment of the CD34- fraction of the graft attempts to address points 1 and 2 (and to a lesser extent point 3) above. We believe that if these points can be consistently achieved that the mortality of allogeneic HSCT may be reduced to levels more akin to those of autologous HSCT. We propose to test the hypothesis that LLME-treated T cells will be safe with regard to reducing GVHD or other infusion related toxicities and that their administration as part of the transplant will facilitate engraftment. We believe that this approach will ultimately be an important step in a variety of transplant settings ranging from matched siblings to haplodisparate donors.