  In this paper we study a normalised anisotropic Gauss curvature flow of
strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove
that the flow exists for all time and converges smoothly to the unique,
strictly convex solution of a Monge-Amp`ere type equation. Our argument
provides a parabolic proof in the smooth category for the existence of
solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.
