The electronic structure of domain walls in superconductors is computed as a function of temperature within Bogoliubov-de Gennes theory. We find a spontaneous interface transition at an ultralow temperature T_* 10^-2 T _c , in which the self-consistent pair potential transforms from a smooth Ginzburg-Landau form into one showing oscillatory Friedel-like behaviour. The transition is associated with a lifting of the energy of midgap excitations and the formation of an energy minigap. Like the bulk superconducting phase transition at T _c , this midgap transition is manifested as a pronounced peak in the specific heat of the domain-wall matter. The transition is also predicted to occur in thin SNS (superconducting-normal-superconducting) Josephson -junctions.