  We study the Stefan problem with surface tension and radially symmetric
initial data. In this context, the notion of a so-called physical solution,
which exists globally despite the inherent blow-ups of the melting rate, has
been recently introduced in [NS23]. The paper at hand is devoted to the proof
that the physical solution is unique, the first such result when the free
boundary is not flat, or when two phases are present. The main argument relies
on a detailed analysis of the hitting probabilities for a three-dimensional
Brownian motion, as well as on a novel convexity property of the free boundary
obtained by comparison techniques. In the course of the proof, we establish a
wide variety of regularity estimates for the free boundary and for the
temperature function, of interest in their own right.
