The purpose of this study is to see if it is safe to give C4-V3, a possible HIV vaccine, alone or in conjunction with 4 different doses of interleukin-12 (IL-12), to HIV-infected patients who are taking anti-HIV drugs that have lowered the amount of HIV in patients' blood. (This study has been changed so that vaccine is administered alone or with 4 different doses of IL-12.) Immune cells known as cytotoxic T lymphocytes (CTLs) help destroy HIV-infected cells. However, in most patients, CTLs decrease over time. This allows HIV levels to rise and AIDS symptoms to develop. The C4-V3 vaccine contains small pieces of HIV protein that can boost CTL levels, allowing the body's immune system to fight HIV. Giving IL-12, a normal part of the immune system, with C4-V3 may make the vaccine more effective.