  In this paper, we study the Severi variety $V_{L,g}$ of genus $g$ curves in
$|L|$ on a general polarized K3 surface $(X,L)$. We show that the closure of
every component of $V_{L,g}$ contains a component of $V_{L,g-1}$. As a
consequence, we see that the general members of every component of $V_{L,g}$
are nodal.
