  In this paper we are interested in a rigorous derivation of the
Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm,
we consider a two-dimensional Stefan problem in a strip, a simplified version
of a solid-liquid interface model. Near the instability threshold, we introduce
a small parameter $\varepsilon$ and define rescaled variables accordingly. At
fixed $\varepsilon$, our method is based on: definition of a suitable linear 1D
operator, projection with respect to the longitudinal coordinate only,
Lyapunov-Schmidt method. As a solvability condition, we derive a
self-consistent parabolic equation for the front. We prove that, starting from
the same configuration, the latter remains close to the solution of K--S on a
fixed time interval, uniformly in $\varepsilon$ sufficiently small.
