An HIV infection dynamics model with intracellular delay is studied. The basic reproduction ratio is obtained by calculation. By analyzing the distribution of roots of the corresponding characteristic equations, the local stability of each of feasible equilibria is established. By constructing the suitable Lyapunov functional, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. From the persistence theory of infinite dimensional dynamic system, it shows that if the basic reproduction ratio is greater than unity, the system is uniformly persistent. According to the iteration technique and comparision arguments, the sufficient conditions are obtained for the global attractivity of the chronic infection equilibrium.