  We prove that the gonality among the smooth curves in a complete linear
system on a $K3$ surface is constant except for the Donagi-Morrison example.
This was proved by Ciliberto and Pareschi under the additional condition that
the linear system is ample.
  As a consequence we prove that exceptional curves on $K3$ surfaces satisfy
the Eisenbud-Lange-Martens-Schreyer conjecture and explicitly describe such
curves. They turn out to be natural extensions of the
Eisenbud-Lange-Martens-Schreyer examples of exceptional curves on $K3$
surfaces.
