The purpose of this study is to optimize blood stem cell mobilization in adults with beta thalassemia major. We seek a method of mobilization that will be safe, with minimum side effects, and that will yield high numbers of blood stem cells. For successful gene therapy of thalassemia, high numbers of genetically modified stem cells will need to be introduced into the patient. Participants will include beta-thalassemia patients who failed to mobilize sufficiently with G-CSF (in our previous protocol) and new patients. In this study we will focus on the safety and effectiveness of mobilization with Mozobil or with Mozobil plus G-CSF. Following mobilization, blood stem cells will be recovered using leukapheresis, a procedure similar to a blood donation, in which mobilized white blood cells are collected from the blood of the patient.