We　investigate　the　stability　of　the　Rayleigh–Taylor　(RT)　problem　for　stratified　viscoelastic　fluids　with　internal　surface　tension.　More　precisely,　under　the　stability　condition　Dis(ϑ,κ)＜1,　we　prove　the　existence　of　unique　strong　solution　with　exponential　decay　in　time　for　the　(stratified)　viscoelastic　RT　(VRT)　problem　with　proper　initial　data　in　Lagrangian　coordinates.　This　shows　that　a　sufficiently　large　elasticity　coefficient　or　a　sufficiently　large　surface　tension　coefficient　has　a　stabilizing　effect　so　that　it　can　inhibit　the　development　of　(stratified)　RT　instability.　©　2020　Elsevier　Ltd