  Let $(S,L)$ be a general primitively polarized $K3$ surface of genus $g$. For
every $0\leq \delta \leq g$ we consider the Severi variety parametrizing
integral curves in $|L|$ with exactly $\delta$ nodes as singularities. We prove
that its closure in $|L|$ is connected as soon as $\delta\leq g-1$. If
$\delta\leq g-4$, we obtain the stronger result that the Severi variety is
irreducible, as predicted by a well-known conjecture. The results are obtained
by degeneration to Halphen surfaces.
