This paper presents an enthalpy based full-state feedback control law with respect to a reference solution for a one-phase Stefan problem under unknown boundary input hysteresis. The one-phase Stefan problem describes the evolution of the temperature and the liquid-solid interface location in a solidifying material. In this paper, this setting is used to model an industrial continuous casting process, which produces nearly all steel currently used worldwide. Regulation of both the steel temperature and the liquid-solid interface location history is the key to the steel quality. Experiments have revealed the existence of hysteresis due to boiling of the cooling water at the surface of the outer (solid) boundary of the solidifying steal shell. This work addresses this difficulty by considering control of the Stefan problem with unknown boundary hysteresis. To reduce the problem complexity, the hysteresis effect uncertainty is represented through the changing parameters. Then, the hysteresis inverse is designed and the recalibration method for the hysteresis inverse is proposed. Simulation results are provided, showing that under this setting, both the temperature and the interface location converge to the reference states.