  In this paper we obtain explicit lower bounds for the radius of convergence
of the Painlev\'e expansions of the Korteweg-de-Vries equation around a movable
singularity manifold ${\Cal S}$ in terms of the sup norms of the arbitrary
functions involved. We use this estimate to prove the well-posedness of the
singular Cauchy problem on ${\Cal S}$ in the form of continuous dependence of
the meromorphic solution on the arbitrary data.
