Umbilical Cord Blood Transplant for Hematological Malignancies

This protocol will enroll subjects with advanced hematologic malignancies who do not have a suitable related or unrelated donor to undergo a Stem Cell Transplant. In this study, subjects will undergo a Stem Cell Transplant using Cord Blood. Part of the cord blood will be used for the Stem Cell Transplant and part of the cord blood will be sent to a laboratory in order to grow the T cells (from the cord blood) and increase the activity of the cord blood T cells. The purpose of this part of the study is to see if it is safe to give study subjects activated T cells made from a small portion of their donor UCB unit immediately after the UCB transplant. Activated T cells have been used safely in stem cell transplantation studies in the past, but they have never been studied UCB transplantation.