We establish a connection between two previously unrelated topics: a particular
discrete version of conformal geometry for triangulated surfaces, and the geometry of
ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are
considered discretely conformally equivalent if the edge lengths are related by
scale factors associated with the vertices. This simple definition leads to a
surprisingly rich theory featuring Möbius invariance, the definition of discrete
conformal maps as circumcircle-preserving piecewise projective maps, and
two variational principles. We show how literally the same theory can be
reinterpreted to address the problem of constructing an ideal hyperbolic
polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a
companion theory of discrete conformal maps for hyperbolic triangulations. It also
shows how the definitions of discrete conformality considered here are closely
related to the established definition of discrete conformality in terms of circle
packings.