This study will last at least 24 weeks and has two steps. Patients will be stratified by HIV viral load and their neurocognitive state at study screening. In Step I, patients will be randomly assigned to one of two groups. Group 1 participants will receive twice-daily minocycline for 24 weeks; Group 2 participants will receive placebo. At the end of Phase I, study participants will be offered to enter Step II; all participants in Step II will receive twice-daily minocycline for an additional 24 weeks.