  We prove that convex hypersurfaces in ${\mathbb R}^{n+1}$ contracting under
the flow by any power $\alpha>\frac{1}{n+2}$ of the Gauss curvature converge
(after rescaling to fixed volume) to a limit which is a smooth, uniformly
convex self-similar contracting solution of the flow. Under additional central
symmetry of the initial body we prove that the limit is the round sphere.
