Dolutegravir (DTG) (also known as GSK1349572) is a new drug that may slow HIV replication. It works by blocking a protein named integrase, which HIV needs in order to make copies of itself in the human body. This drug has been tested for safety and effectiveness in adults but not in children. This study will test the safety of and immune response to DTG in HIV-1 infected infants, children, and adolescents.