Abstract: The one-dimensional one-phase Supercooled Stefan Problem is a PDE problem with free boundary which serves as a model for the freezing of supercooled liquids. Under certain conditions, this model will exhibit blow-up in finite time. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the Supercooled Stefan Problem through the Fokker-Planck equation associated to a stochastic process that solves a certain McKean-Vlasov equation. This technique allows us to define solutions globally even in the presence of blow-ups. Solutions to the associated McKean-Vlasov equation can be constructed via an approximating particle system, and we prove Propagation of Chaos. The particle system in question appears in the literature on systemic risk, establishing the connection of the aforementioned results to Mathematical Finance. Finally, we prove a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.