Analysed is a mathematical model for HIV-1 infection with two delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells. For this model, the basic reproduction number is identified and its threshold property is discussed: the uninfected steady state is proved to be globally asymptotically stable if and unstable if . In the latter case, an infected steady state occurs and is proved to be locally asymptotically stable. The formula for shows that increasing either of the two delays will decrease . This may suggest a new direction for new drugs—drugs that can prolong the latent period and/or slow down the virus production process.