In this paper, we present a new network design problem that is applicable for designing virtual paths (VP's) in an asynchronous transfer mode (ATM) network to efficiently support multicast applications, especially real-time multimedia applications. We first address several alternatives for the solution and compare their properties. Then we focus on a new solution which sets up a semi-permanent VP layout (VPL) and constructs VC trees for different multicast traffic demand patterns based on the constructed VPL. A three-phase heuristic solution is proposed for designing a good virtual-path layout for a given set of multicast traffic demand patterns. By varying the design parameters, we can obtain different VPLs which possess different tradeoffs among some important criteria, namely, the network overhead for a connection setup, routing table resources and control and management cost. Simulations are performed on randomly generated networks to demonstrate the performance and scalability of our solution. To the best of our knowledge, there is no prior known work which takes the multicast connection traffic into account for the VP layout design.