 This study will last approximately 31 months. Participants will be randomly assigned to one of three groups at study entry. Group A will receive no aldesleukin or HAART. Group B will receive aldesleukin under the skin twice daily for 5 consecutive days every 8 weeks for 3 cycles, then as needed to maintain CD4 counts at or above a goal level. Group C will receive aldesleukin under the skin twice daily for 5 consecutive days every 8 weeks for 3 cycles, then as needed to maintain CD4 counts at or above a goal level; Group C participants will also take HAART for 3 days prior to the start of each aldesleukin cycle, throughout the 5-day aldesleukin cycle, and for 2 days after the end of each aldesleukin cycle for a maximum of 10 days with each aldesleukin cycle. HAART will not be provided by the study. Some Group B and C participants may take part in additional cycles of aldesleukin if they meet certain study criteria. 