An ATM network design algorithm is treated as a resource allocation problem. As an effective way to facilitate a coexistence of traffic with its diverse characteristics and different quality of service (QOS) requirements in ATM networks, a virtual path (VP) concept has been proposed. In attempting to design the VP (Virtual Path)-based ATM network, it requires to consider a network topology and traffic pattern generated from users for minimizing a network construction cost while satisfying QOS requirements such as cell / call loss probabilities and cell delay times. In this paper, we propose a new heuristic design algorithm for the VP-based ATM network under QOS constraints. A minimum bandwidth required to transfer a given amount of traffic is first obtained by utilizing an equivalent bandwidth method. After all the routes of VPs are temporarily established by means of the shortest paths, we try to minimize the network cost through the alternation of VP route, the separation of a single VP into several VPs, and the introduction of VCX nodes. To evaluate our design algorithm, we consider two kinds of traffic; voice traffic as low speed service and still picture traffic as high speed service. Through numerical examples, we demonstrate that our design method can achieve an efficient use of network resources, which results in the cost-effective VP-based ATM network.