  We develop a framework for a unified treatment of well-posedness for the
Stefan problem with or without surface tension. In the absence of surface
tension, we establish well-posedness in Sobolev spaces for the classical Stefan
problem. We introduce a new velocity variable which extends the velocity of the
moving free-boundary into the interior domain. The equation satisfied by this
velocity is used for the analysis in place of the heat equation satisfied by
the temperature. Solutions to the classical Stefan problem are then constructed
as the limit of solutions to a carefully chosen sequence of approximations to
the velocity equation, in which the moving free-boundary is regularized and the
boundary condition is modified in a such a way as to preserve the basic
nonlinear structure of the original problem. With our methodology, we
simultaneously find the required stability condition for well-posedness and
obtain new estimates for the regularity of the moving free-boundary. Finally,
we prove that solutions of the Stefan problem with positive surface tension
$\sigma$ converge to solutions of the classical Stefan problem as $\sigma \to
0$.
