  We study the solutions of the one-phase supercooled Stefan problem with
kinetic undercooling, which describes the freezing of a supercooled liquid, in
one spatial dimension. Assuming that the initial temperature lies between the
equilibrium freezing point and the characteristic invariant temperature
throughout the liquid our main theorem shows that, as the kinetic undercooling
parameter tends to zero, the free boundary converges to the (possibly
irregular) free boundary in the supercooled Stefan problem without kinetic
undercooling, whose uniqueness has been recently established in [DNS19],
[LS18b]. The key tools in the proof are a Feynman-Kac formula, which expresses
the free boundary in the problem with kinetic undercooling through a local time
of a reflected process, and a resulting comparison principle for the free
boundaries with different kinetic undercooling parameters.
