  Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped
with a complete hyperbolic cusp metric. Then it can be uniquely realized as the
boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a
new variational proof of this result. We also give an alternative proof of the
existence and uniqueness of a hyperbolic polyhedral metric with prescribed
curvature in a given conformal class.
