  We address the asymptotic behavior of the $\alpha$-Gauss curvature flow, for
$\alpha >1/2$, with initial data a complete non-compact convex hypersurface
which is contained in a cylinder of bounded cross section. We show that the
flow converges, as $t \to +\infty$, locally smoothly to a translating soliton
which is uniquely determined by the asymptotic cylinder of the initial
hypersurface.
