The purpose of this study is to determine the effectiveness of a new treatment for multiple sclerosis (MS), a serious disease in which the immune system attacks the brain and spinal cord. MS can be progressive and severe and lead to significant disability. The study treatment involves the use of high-dose chemotherapeutic drugs to suppress the immune system. The participant's own (autologous) blood-forming (hematopoietic, CD34+) stem cells are collected before the chemotherapy is given, and then transplanted back into the body following treatment. Transplantation of autologous hematopoietic stem cells is required to prevent very prolonged periods of low blood cell counts after the high-dose chemotherapy.