We study a survivable network design problem:the directed network design problem with connectivity constraints (DNCC). Some applications in telecommunications are presented. We discuss two integer linear programming models for DNCC, and relate these. The main body of the paper is a study of DNCC from a polyhedral point of view. We give several classes of nonredundant inequalities for polytopes associated with the problem. A cutting plane algorithm based on the polyhedral results is described and some computational results are given.