This paper focuses on the boundary conditions which are applied at the boundary between a melt (i.e., a molten binary alloy) and adjacent mush (region of mixed liquid and solid which forms by partial solidification of the melt). It is generally believed that continuity of variables and fluxes leaves the problem underdetermined and an auxiliary condition must arbitrarily be specified at this interface. The two alternatives which have been proposed are marginal phase equilibrium and zero solid fraction. It is shown that the underdeterminacy may be resolved by application of an inequality assuring that the melt is not supercooled, together with the condition that the mass fraction of solid have physically realistic values. The lack of supercooling is a natural corollary of the assumption of thermodynamic phase equilibrium in the mush. For a very wide range of parameter values, this corollary leads directly to the "extra" boundary conditions that have been previously proposed: marginal phase equilibrium and zero solid fraction; no auxiliary condition need be applied. Phase disequilibrium does not alter the conclusion that the solid fraction at the boundary must be zero.