  We formulate a Stefan problem on an evolving hypersurface and study the
well-posedness of weak solutions given $L^1$ data. To do this, we first develop
function spaces and results to handle equations on evolving surfaces in order
to give a natural treatment of the problem. Then we consider the existence of
solutions for $L^\infty$ data; this is done by regularisation of the
nonlinearity. The regularised problem is solved by a fixed point theorem and
then uniform estimates are obtained in order to pass to the limit. By using a
duality method we show continuous dependence which allows us to extend the
results to $L^1$ data.
