Patients will have MRI scans before each infusion. The MRIs will be done using the standard procedure and again using a contrast agent, gadolinium, injected into a vein. Gadolinium helps identify new multiple sclerosis lesions in the brain. Blood and urine samples will be taken during each clinic visit. In addition, patients will have skin tests, similar to a tuberculin test, to evaluate immune status, and will be asked to undergo two lumbar punctures (spinal tap; these will be optional)-one before the treatment phase begins, and another when treatment is completed. Lymphocytes will also be collected from patients before, during and after treatment. The lymphocytes are obtained by a procedure called apheresis: about a pint of whole blood is drawn through a needle in the arm, the lymphocytes are separated out and removed by a machine, and the rest of the blood is returned through a needle in the other arm. These studies will hopefully allow conclusions about the safety of Zenapax in MS, but also address its effectiveness with respect to modifying the inflammatory activity in the brain of MS patients and inhibit autoimmune T lymphocytes that are involved in the disease process.