  Let $\mathbb Q^{n+1}_c$ be the complete simply-connected $(n+1)$-dimensional
space form of curvature $c$. In this paper we obtain a new characterization of
geodesic spheres in $\mathbb Q^{n+1}_c$ in terms of the higher order mean
curvatures. In particular, we prove that the geodesic sphere is the only
complete bounded immersed hypersurface in $\mathbb Q^{n+1}_c,\;c\leq 0,$ with
constant mean curvature and constant scalar curvature. The proof relies on the
well known Omori-Yau maximum principle, a formula of Walter for the Laplacian
of the $r$-th mean curvature of a hypersurface in a space form, and a classical
inequality of G\aa rding for hyperbolic polynomials.
