This paper deals with a Steiner tree-star problem that is a special case of the degree constrained node-weighted Steiner tree problem. This problem arises in the context of designing telecommunications networks for digital data service, provided by regional telephone companies. In this paper, we develop an effective branch and cut procedure coupled with a suitable separation procedure that identifies violated facets for fractional solutions to the relaxed formulation. Computational results indicate that large problem instances with up to 200 nodes can be solved within acceptable time bounds.