Computer networks are undergoing a remarkable transformation. The widespread use of optical fiber has tremendously increased network bandwidth. Furthermore, greater CPU power, increasing disk capacity, and support for digital audio and video are creating demand for a new class of network services. For example, video-on-demand (VOD), distant learning, distant diagnosis, video conferences, and many other applications have popped up one after another in recent years. Many of these services have one thing in common. They all require that the same data be sent to multiple recipients. Even in traditional networks, this operation, called multicasting, can not be handled easily and cheaply. When scaled up to high-speed ATM-based networks, the situation becomes more difficult. Multiple data streams travel around the ATM networks with each stream sent to many different destinations simultaneously. Therefore, the networks need a very efficient algorithm for multicast routing. This paper proposes an algorithm for multicast routing in ATM networks. The network routing cost is defined as the combination of the cost of establishing connections, the cost of overall bandwidth and the cost of overall switching. Our routing algorithm generates an optimal multicast routing with minimal overall network cost. The widespread use of optical fiber has tremendously increased network bandwidth. Furthermore, greater CPU power, increasing disk capacity, and support for digital audio and video are creating demand for a new class of network services. For example , video-on-demand (VOD), distant learning, distant diagnosis, video conferences, and many other applications have popped up one after another in recent years. Many of these services have one thing in common. They all require that the same data Even in traditional networks, this operation, called multicasting, can not be handled easily and cheaply. When scaled up to high-speed ATM-based networks, the situation becomes more difficult. Multiple data streams travel around the ATM networks with each stream sent to many different destinations simultaneously. This, the networks need a very efficient algorithm for multicast routing an algorithm for multicast routing in ATM networks. The network routing cost is defined as the combination of the cost of establishing connections, the cost of overall bandwidth and the cost of overall switching. Our routing algorithm generates an optimal multicast routing with minimal overall network cost .