Locally convex hypersurfaces immersed in $H^n times R$

We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to $R^n$. In the latter case it is either a vertical graph over a convex domain in $H^n$ or has what we call a simple end.