This paper is concerned with a restricted version of minimum cost delay-constrained multicast in a network where each link has a delay and a cost. Given a source vertex ss and pp destination vertices t₁, t₂, ¼, t_pt_1, t_2, \ldots, t_p together with pp corresponding nonnegative delay constraints d₁, d₂, ¼, d_pd_1, d_2, \ldots, d_p, many QoS multicast problems seek a minimum cost multicast tree in which the delay along the unique ss--t_it_i path is no more than d_id_i for 1 £ i £ p1 \le i \le p. This problem is NP-hard even when the topology of the multicast tree is fixed. In this paper we show that every multicast tree has an underlying Steiner topology and that every minimum cost delay-constrained multicast tree corresponds to a minimum cost delay-constrained realization of a corresponding Steiner topology. We present a fully polynomial time approximation scheme for computing a minimum cost delay-constrained multicast tree under a Steiner topology. We also present computational results of a preliminary implementation to illustrate the effectiveness of our algorithm and discuss its applications.