This paper  discretizes the SIR model with a saturated incidence rate using a nonstandard finite difference method, then solves and analyzes the properties of the  discrete SIR model. The existence positivity and boundedness of the model's solution are obtained. It is shown that the disease-free equilibrium $E_0$ is globally asymptotically stable if the basic reproductive number $R_0<1$, the endemic equilibrium $E^*$ is globally asymptotically stable if $R_0>1$ by constructing a proper V-function. Finally, the numerical simulation supports the proposed theoretical results. According to the simulation, the step length $h$ of the nonstandard finite difference method, rather than that of the forward euler method, has no effect on the global dynamics of the discrete model.