The existence of von Neumann-Morgenstern solutions (stable sets) for assignment games has been an unsolved question since Shapley and Shubik (1972). For each optimal matching between buyers and sellers, Shubik (1984) proposed considering the union of the core of the game and the core of the subgames that are compatible with this matching. We prove in the present paper that this set is the unique stable set for the assignment game that excludes third-party payments with respect to a fixed optimal matching. Moreover, the stable sets that we characterize, as well as any other stable set of the assignment game, have a lattice structure with respect to the same partial order usually defined on the core.