The basic reproductive number R_0 of a discrete SIR epidemic model is defined and the dynamical behavior of the model is studied. It is proved that the disease free equilibrium is globally asymptotically stable if R_0<1, and the persistence of the model is obtained when R_0>1. The main attention is paid to the global stability of the endemic equilibrium. Sufficient conditions for the global stability of the endemic equilibrium are established by using comparison principle. Numerical simulations are done to show our theoretical results and to demonstrate the complicated dynamics of the model.