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The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. Motivated by methods from the Euler equations, we establish a global-in-time stability result for small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Harnack-type inequalities.  This is joint work with M. Hadzic.