We investigate the polyhedral structure of an integer program with a single functional constraint: the integer capacity-cover polyhedron. Such constraints arise in telecommunications planning and facility location applications, and feature the use of general integer (rather than just binary) variables. We derive a large class of facet-defining inequalities by using an augmenting technique that builds upon the facets of a family of related knapsack-cover polyhedra. To the best of our knowledge, this technique is new, and in particular it differs from sequential or simultaneous lifting. It demonstrates an interesting theoretical connection between the facial structure of two families of polyhedra important in applications. Additionally, we derive another class of facet-defining inequalities via coefficient reduction.