  In this paper, we introduce a novel way to represent the interface for
two-phase flows with phase change. We combine a level-set method with a
Cartesian embedded boundary method and take advantage of both. This is part of
an effort to obtain a numerical strategy relying on Cartesian grids allowing
the simulation of complex boundaries with possible change of topology while
retaining a high-order representation of the gradients on the interface and the
capability of properly applying boundary conditions on the interface. This
leads to a two-fluid conservative second-order numerical method. The ability of
the method to correctly solve Stefan problems, onset dendrite growth with and
without anisotropy is demonstrated through a variety of test cases. Finally, we
take advantage of the two-fluid representation to model a Rayleigh--B\'enard
instability with a melting boundary.
