At the beginning of the study, participants will undergo a number of screening and baseline procedures, including a physical exam, blood collection, MS-confirming neurology exams and questionnaires, and MRI procedures. Participants will be given prednisone and granulocyte-colony stimulating factor (G-CSF) to mobilize CD34+ hematopoietic stem cells from the bone marrow into the peripheral blood. When the peripheral blood CD34+ cell count reaches 20,000 cells/ml or greater, these cells will be collected by leukapheresis. In this process, a catheter is placed into a large blood vessel, peripheral blood is withdrawn, and a high speed sedimentation (leukapheresis) device is used to separate and retain the cells required for autologous transplantation. Other blood cells are then returned to the participant's body. In the laboratory, the CD34+ hematopoietic stem cell graft will be selected and prepared from the leukapheresis collection, and stored until needed for transplant. Seven or more days following the collection of their autologous graft, participants will be hospitalized and receive high-dose chemotherapy consisting of carmustine, etoposide, cytarabine, and melphalan (BEAM) and thymoglobulin. This is followed by transplantation of the autologous hematopoietic cell graft. Participants will remain in the hospital for observation during recovery of their peripheral blood cell counts, as described in the protocol. Participants will receive G-CSF and blood transfusions, if needed, and will be monitored for infections. Following discharge from the hospital, eight study visits will occur over sixty months (five years). During these visits, participants will undergo blood and urine collection, MRI studies, leukapheresis, and MS neurology exams and will complete questionnaires.