  We consider the problem of recovering the initial condition in the
one-dimensional one-phase Stefan problem for the heat equation from the
knowledge of the position of the melting point. We first recall some properties
of the free boundary solution. Then we study the uniqueness and stability of
the inversion. The principal contribution of the paper is a new logarithmic
type stability estimate that shows that the inversion may be severely
ill-posed. The proof is based on integral equations representation techniques,
and the unique continuation property for parabolic type solutions. We also
present few numerical examples operating with noisy synthetic data.
