We　investigate　stability　of　an　equilibrium　state　to　a　nonhomogeneous　incompressible　viscoelastic　fluid　driven　by　gravity　in　a　bounded　domain　Ω⊂R3　of　class　C3.　First,　we　establish　a　critical　number　κC,　which　depends　on　the　equilibrium　density　and　the　gravitational　constant,　and　is　a　threshold　of　the　elasticity　coefficient　κ　for　instability　and　stability　of　the　linearized　perturbation　problem　around　the　equilibrium　state.　Then　we　prove　that　the　equilibrium　state　is　exponential　stability　provided　that　κ>κC　and　the　initial　disturbance　quantities　around　the　equilibrium　state　satisfy　some　relations.　In　particular,　if　the　equilibrium　density　ρ-　is　a　Rayleigh-Taylor　(RT)　type　and　ρ-＇　is　a　constant,　our　result　strictly　shows　that　the　sufficiently　large　elasticity　coefficient　can　prevent　the　RT　instability　from　occurrence.　©　2016　Elsevier　Inc.