  In this article we exhibit certain projective degenerations of smooth $K3$
surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the
hyperplane class), to a union of two rational normal scrolls, and also to a
union of planes. As a consequence we prove that the general hyperplane section
of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$.
We also prove that the general such hyperplane section lies on a unique $K3$
surface, up to projectivities. Finally we present a new approach to the
classification of prime Fano threefolds of index one, which does not rely on
the existence of a line.
