  In this paper, we prove that convex hypersurfaces under the flow by powers
$\alpha>0$ of the Gauss curvature in space forms $\mathbb{N}^{n+1}(\kappa)$ of
constant sectional curvature $\kappa$ $(\kappa=\pm 1)$ contract to a point in
finite time $T^*$. Moreover, convex hypersurfaces under the flow by power
$\alpha>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling) to a
limit which is the geodesic sphere in $\mathbb{N}^{n+1}(\kappa)$. This extends
the known results in Euclidean space to space forms.
