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　the　comparison　principle.　Numerical　simulations　are　done　to　show　our　theoretical　results　and　to　demonstrate　the　complicated　dynamics　of　the　model.