Real-time communication networks are designed mainly to support multimedia applications, especially the interactive ones, which require a guarantee of Quality of Service (QoS). Moreover, multicasting is needed as there are usually more than two peers who communicate together using multimedia applications. As for the routing, the network has to find an optimum (least cost) multicast route, that has enough resources to provide or guarantee the required QoS. This problem is called QoS constrained multicast routing and was proved to be an NP-complete problem. In contrast to the existing heuristic approaches, in this paper we propose a modified version of a Hopfield neural network model to solve QoS (delay) constrained multicast routing. By the massive parallel computation of neural networks, it can find a near optimal multicast route very fast, when implemented in hardware. Simulation results show that the proposed model has performance near to the optimal solution and comparable to existing heuristics