It is well known that hypersurfaces with constant mean curvature in Riemannian spaces are solutions to the variational problem of minimizing area for volume-preserving variations. Less known is the fact that hypersurfaces with constant scalar curvature in Riemannian spaces of constant sectional curvature are also solutions to a variational problem, namely, that of minimizing the integral of the mean curvature for volume preserving variations. Thus questions of stability can be posed for hypersurfaces with constant scalar curvature. The following result is proved. Let \(M\) be a compact immersed hypersurface of the euclidean space, or of a sphere, with constant scalar curvature. In the case the ambient space is a sphere, assume, in addition, that \(M\) is contained in an open hemisphere. Then \(M\) is stable if and only if \(M\) is a geodesic sphere. The question envolves the study of a second order differential operator related to the Laplacian and the second fundamental form of the hypersurface.