  We show how the Stefan type free boundary problem with random diffusion in
one space dimension can be approximated by the corresponding free boundary
problem with nonlocal diffusion. The approximation problem is a slightly
modified version of the nonlocal diffusion problem with free boundaries
considered in [4,8]. The proof relies on the introduction of several auxiliary
free boundary problems and constructions of delicate upper and lower solutions
for these problems. As usual, the approximation is achieved by choosing the
kernel function in the nonlocal diffusion term of the form $J_\epsilon(x)=\frac
1\epsilon J(\frac x\epsilon)$ for small $\epsilon>0$, where $J(x)$ has compact
support. We also give an estimate of the error term of the approximation by
some positive power of $\epsilon$.
