The purpose of this study is to look at the effects of the HIV vaccine Remune on viral load (level of HIV in the blood) and on the way the immune system responds to HIV. This study will also try to see if the effects of the vaccine are different in patients entering the study with a viral load below 50 copies/ml compared to those who have a viral load from 50 to 500 copies/ml. (This study is currently being redesigned and the purpose may be revised.) Treatment with anti-HIV drugs does not always keep HIV viral load undetectable (so low that it cannot be measured). This study originally added an HIV vaccine called Remune to treat patients. Remune was thought to reduce viral load and improve immune responses. However, new information suggests that Remune may not be as effective as was first believed. The study has been changed to follow people already in the study and to let people enroll only if they participate in the substudy. The substudy will look at the effect of another HIV vaccine, vCP1452, on the immune response and how it works in combination with Remune. Information about the safety of these vaccines in HIV-positive patients will be gathered.