This study will examine the safety and effectiveness of a new drug combination for treating patients with severe aplastic anemia. Patients with aplastic anemia produce too few blood cells, causing fatigue, easy bruising and bleeding, and susceptibility to infections. In many cases, the very low blood counts result from an autoimmune process-that is, the patient's own immune system suppresses production of blood cells by the bone marrow. Although immune-suppressing drugs, such as cyclosporine, can restore normal cell counts, many patients have disease relapses. These patients require long-term therapy with cyclosporine, which can cause harmful side effects. This study will examine whether a lower dose of cyclosporine given together with mycophenolate mofetil (MMF) can maintain blood counts as effectively as full-dose cyclosporine treatment, and whether MMF alone can reduce the chances of future relapses.