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.