  This paper deals with the exact controllability to the trajectories of the
one--phase Stefan problem in one spatial dimension. This is a free-boundary
problem that models solidification and melting processes. It is assumed that
the physical domain is filled by a medium whose state is liquid on the left and
solid, with constant temperature, on the right. In between we find a
free-boundary (the interface that separates the liquid from the solid). In the
liquid domain, a parabolic equation completed with initial and boundary
conditions must be satisfied by the temperature. On the interface, an
additional free-boundary requirement, called the {\it Stefan condition,} is
imposed. We prove the local exact controllability to the (smooth) trajectories.
To this purpose, we first reformulate the problem as the local null
controllability of a coupled PDE-ODE system with distributed controls. Then, a
new Carleman inequality for the adjoint of the linearized PDE-ODE system,
coupled on the boundary through nonlocal in space and memory terms, is
presented. This leads to the null controllability of an appropriate linear
system. Finally, a local result is obtained via local inversion, by using {\it
Liusternik-Graves' Theorem}. As a byproduct of our approach, we find that some
parabolic equations which contains memory terms localized on the boundary are
null-controllable.
