  Let $(S,L)$ be a polarized $K3$ surface of genus $p \geqslant 11$ such that
$\mathrm{Pic}(S)=\mathbf{Z}[L]$, and $\delta$ a non-negative integer. We prove
that if $p\geqslant 4\delta-3$, then the Severi variety of $\delta$-nodal
curves in $|L|$ is irreducible.
