We　choose　a　constitutive　model,　which　consists　of　a　viscous　stress　component　and　a　stress　component　for　a　neo-Hookean　solid,　to　describe　the　motion　of　a　viscoelastic　fluid　heated　from　below,　and　then　mathematically　investigate　the　stability　for　the　Rayleigh-Bénard　problem　of　the　constitutive　model.　A　stability　criterion　is　established,　under　which　the　Rayleigh-Bénard　problem　is　exponentially　stable　with　respect　to　time.　Our　stability　result　shows　that　the　elasticity　can　inhibit　the　thermal　instability　under　sufficiently　large　elasticity　coefficient　k.　In　addition,　we　also　provide　an　instability　criterion,　under　which　the　Rayleigh-Bénard　problem　is　unstable.　Our　instability　result　shows　that　elasticity　can　not　inhibit　the　thermal　instability,　when　k　is　too　small.　©　2020　IOP　Publishing　Ltd　&　London　Mathematical　Society.