This paper presents an observer and an output feedback control with respect to a reference solution for the one-phase Stefan problem under input hysteresis. The one-phase Stefan problem describes evolution of the temperature and melting-solidification front in liquid-solid material. The setting models an industrial casting process, and experiments have revealed the existence of hysteresis due to boiling of the cooling water at the surface of the casting process. Therefore, one-phase Stefan problem with water cooling hysteresis under Neumann boundary actuation is considered. Full state feedback control law for this problem was designed and proved to provide asymptotic convergence of both temperature distribution and the solidification front. However, for the casting process, only boundary sensing is available. To address the latter problem, the present paper proposes an observer that estimates the temperature profile based on the available surface temperature measurement, taking into account boundary input hysteresis. The stability of the observer is proved with Lypanov method. Finally, an output feedback control law is proposed and proved to ensure asymptotic convergence of the temperature and the solidification front errors to zero. A numerical example presents the application of the method proposed.