Patients with severe aplastic anemia who do not have a suitable bone marrow donor or who decline bone marrow transplantation may participate in this study. Patients will have a skin test for ATG allergy, chest X-ray, blood test, and bone marrow aspiration before treatment begins. ATG will then be started, infused through a vein continuously for 4 days. Ten days after ATG is stopped, cyclosporine treatment will begin, taken twice a day by mouth in either liquid or capsule form and will continue for 6 months. Also, in the first 2 weeks of treatment, patients will be given a full dose of corticosteroid (prednisone) to prevent serum sickness that could develop as a side effect of ATG therapy. The dosage will be decreased after that. Mycophenolate will be started at the same time as ATG, in two daily doses by mouth, and will continue for 18 months.