We construct for every connected surface $S$ of finite negative Euler characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N$ of finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to N$ 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. Joint work with Colin Adams and William Meeks.