  In this paper we study the Gauss map of hypersurfaces with constant weighted
mean curvature in the Gaussian space. We show that if the image of the Gauss
map is in a closed hemisphere, then the hypersurface is a hyperplane or a
generalized cylinder. We also show that if the image of the Gauss map is in
$S^{n}\setminus\bar{S}_{+}^{n-1}$, then the hypersurface is a hyperplane. This
generalizes previous results for self-shrinkers obtained by Ding-Xin-Yang.
