We use a variational principle to prove an existence and uniqueness theorem for
planar weighted Delaunay triangulations (with non-intersecting site-circles)
with prescribed combinatorial type and circle intersection angles. Such weighted
Delaunay triangulations may be interpreted as images of hyperbolic polyhedra
with one vertex on and the remaining vertices beyond the infinite boundary of
hyperbolic space. Thus, the main theorem states necessary and sufficient
conditions for the existence and uniqueness of such polyhedra with prescribed
combinatorial type and dihedral angles. More generally, we consider weighted
Delaunay triangulations in piecewise flat surfaces, allowing cone singularities
with prescribed cone angles in the vertices. The material presented here extends
work by Rivin on Delaunay triangulations and ideal polyhedra.