  Let $M^n$ be either a simply connected space form or a rank-one symmetric
space of noncompact type. We consider Weingarten hypersurfaces of
$M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$
and angle function $\theta$ satisfy a relation $W(k_1,\dots,k_n,\theta^2)=0,$
being $W$ a differentiable function which is symmetric with respect to
$k_1,\dots, k_n.$ When $\partial W/\partial k_i>0$ on the positive cone of
$\mathbb R^n,$ a strictly convex Weingarten hypersurface determined by $W$ is
said to be elliptic. We show that, for a certain class of Weingarten functions
$W,$ there exist rotational strictly convex Weingarten hypersurfaces of
$M\times\mathbb R$ which are either topological spheres or entire graphs over
$M.$ We establish a Jellett-Liebmann-type theorem by showing that a compact,
connected and elliptic Weingarten hypersurface of either $\mathbb
S^n\times\mathbb R$ or $\mathbb H^n\times\mathbb R$ is a rotational embedded
sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces
of these ambient spaces are obtained. We also obtain existence results for
constant scalar curvature hypersurfaces of $\mathbb S^n\times\mathbb R$ and
$\mathbb H^n\times\mathbb R$ which are either rotational or invariant by
translations (parabolic or hyperbolic). We apply our methods to give new proofs
of the main results by Manfio and Tojeiro on the classification of constant
sectional curvature hypersurfaces of $\mathbb S^n\times\mathbb R$ and $\mathbb
H^n\times\mathbb R.$
