  We examine travelling wave solutions of the reaction-diffusion equation,
$\partial_t u= R(u) + \partial_x \left[D(u) \partial_x u\right]$, with a
Stefan-like condition at the edge of the moving front. With only a few
assumptions on $R(u)$ and $D(u)$, a variety of new compactly supported
travelling waves arise in this Reaction-Diffusion Stefan model. While other
reaction-diffusion models admit compactly supported travelling waves for a
unique wavespeed, we show that compactly supported travelling waves in the
Reaction-Diffusion Stefan model exist over a range of wavespeeds. Furthermore,
we determine the necessary conditions on $R(u)$ and $D(u)$ for which compactly
supported travelling waves exist for all wavespeeds. Using asymptotic analysis
in various distinguished limits of the wavespeed, we obtain approximate
solutions of these travelling waves, agreeing with numerical simulations with
high accuracy.
