This study will test the safety and effectiveness of a combination of three drugs in treating severe aplastic anemia and preventing its recurrence. Two drugs used in this trial ATG and cyclosporine are standard combination therapy for aplastic anemia. This study will try to improve this therapy in three ways: 1) by altering the drug regimen to allow the drugs to work better; 2) by reducing the risk of kidney damage; and 3) by adding a third drug mycophenolate mofetil to try to prevent disease relapse.