In　this　paper,　we　consider　a　Leslie-Gower　model　with　weak　Allee　effect　in　the　prey.　By　analysing　the　dynamics　near　the　origin,　we　show　that　both　predator　and　prey　will　tend　to　extinction　if　the　intensity　of　Allee　effect　is　strong　enough.　Meanwhile,　we　provide　some　sufficient　conditions　on　the　global　asymptotic　stability　of　the　unique　positive　equilibrium.　In　addition,　Allee　effect　can　change　the　stability　of　positive　equilibrium,　which　leads　to　the　occurrence　of　a　supercritical　Hopf　bifurcation　and　one　stable　limit　cycle.　It　is　interesting　to　note　that　there　exists　at　least　one　limit　cycle　around　the　unstable　positive　equilibrium.　In　particular,　sufficient　conditions　for　the　existence　of　a　unique　stable　limit　cycle　have　been　presented.　Numerical　simulations　are　conducted　to　validate　the　main　results.