Growth hormone plays an important role in the development of the immune system. Studies suggest that growth hormone may promote growth of the thymus, a gland responsible for the production of important immune cells called T cells. Since these cells are lost during the course of HIV infection, it is possible that growth hormone treatment could help restore the immune system. This study will determine whether the administration of growth hormone can increase the size and function of the thymus and cause an increase in the number of new T cells in the blood of people infected with HIV.