  We construct for every connected surface $S$ of finite negative Euler
characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N(S,H)$ of
finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to
N(S,H)$ with mean curvature $H$. Conversely, we prove that a complete, totally
umbilic surface with mean curvature $H \in [0,1)$ embedded in a hyperbolic
3-manifold of finite volume must be proper and have finite negative Euler
characteristic.
