In this paepr, we propose a dynamic multicast routin algorithm for constructing the delay-constrained minimal spanning tree in the VP-based ATM networks, in which we consider the effiiciency enen in the case wheree the destination dynamically joins/departs the multicast connection. For constructing the delay-constrained spanning tree, we frist generate a reduced network consisting of only VCX nodes from a given ATM network, originally consisting of VPX/VCX nodes. Then, we obtain the delay-constrained spanning tree with a minimal tree cost on the reduced network by using our proposed heuristic algorithm. Through numerical examples, we show that our dynamic multicast routing algorithm can provide an efficient usage of network resources when the membership nodes frequently changes during the lifetime of a multicast connection. We also demonstrate the more cost-saving can be expected in dense networks when applyingour proposed algorithm.