This article concerns the structure of complete noncompact stable hypersurfaces M n with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N n+1 . In particular, we show that a complete noncompact stable constant mean curvature hypersurface M n , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with H 2 > 65 63 , 175 148 , 41 25 , 671 171 for n = 3, 4, 5, 6, respectively, has only one end.