 All participants in this study will have at least 8 study visits; these visits will occur at study entry and Weeks 4, 8, 12, 16, 20, 24, and 32. Blood collection will occur at all visits and will include tests for CD4 count and viral load. Groups B and C will have additional blood collection within 4 days prior to the start of each aldesleukin cycle. On the last day of each aldesleukin cycle, Groups B and C will be assessed for toxicities, adverse events, and adherence to the aldesleukin daily injections; they will also have another blood collection. Group C participants will have an additional blood collection for HIV genotyping after they have completed their third aldesleukin cycle. Extended follow-up visits will occur approximately every 4 months for an additional two years. Blood collection will occur at these visits and will include tests for CD4 count, viral load, and other laboratory tests. 