We investigate the uniqueness of stable coalition structures, when the value of a coalition to a member depends solely on the identity of the other members of the coalition. We give a complete characterization of collections of coalitions which ensure that there is a unique stable coalition structure at every preference profile when only admissible coalitions may form. An alternative characterization of unique stability is also provided, based on a graph- theoretic representation of collections of coalitions. We explore the implications of our characterization results for matching problems and the existence of strategyproof rules of coalition formation.