  We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary
is uniquely determined by the metric induced on its boundary. Furthemore, any
hyperbolic metric on the torus with cone singularities of positive curvature
can be realized as the induced metric on the boundary of a convex polyhedral
cusp. The proof uses the total scalar curvature functional on the space of
``cusps with particles'', which are hyperbolic cone-manifolds with the singular
locus a union of half-lines. We prove, in addition, that convex polyhedral
cusps with particles are rigid with respect to the induced metric on the
boundary and the curvatures of the singular locus. Our main theorem is
equivalent to a part of a general statement about isometric immersions of
compact surfaces.
