In this paper, we propose a new multicast routing algorithm for constructing the delay-constrained minimal spanning tree in the VP-based ATM networks, in which we consider the efficiency even in the case where the destination dynamically joins/departs the multicast connection. For constructing the delay-constrained spanning tree, we first 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, while the existing multicast routing algorithm may be useful for constructing the multicast tree with a static nature of destination nodes. We also demonstrate that more cost-saving can be expected in dense networks when applying our proposed algorithm.