In this paper we propose a distributed protocol for constructing delay-bounded minimum-cost multicast trees in computer networks. The proposed protocol does not require knowledge of the complete network topology. Furthermore, it is unnecessary to know how many nodes are in the networks and which nodes are group members in advance. It is shown, through analysis and simulation on a class of random graphs, that our protocol only uses O(n) messages in the best case, where n is the number of nodes in the networks. Even in the worst case, our protocol uses O(d*n) messages that has lower message complexity than previous protocols, where d is the average nodal degree.