The delivery of multimedia contents through a Content Delivery Network (CDN) is typically handled by a specific third party, separated from the content provider. However, in some specific cases, the content provider may be interested in carrying out this function using a Private CDN, possibly using an off-sourced network infrastructure. This scenario poses new challenges and limitations with respect to the typical case of content delivery. First, the systems has to face a different workload as the content consumer are typically part of the same organization that is the content provider. Second, the offsourced nature of the network infrastructure has a major impact on the available choices for CDN design. In this paper we develop an exact mathematical model for the design of a Private CDN addressing the issues and the constraints typical of such scenario. Furthermore, we analyze different heuristics to solve the optimization problem. We apply the proposed model to a real case study and validate the results by means of simulation.