  We show that a small variant of the methods used by Voisin in her study of
canonical curves leads to a surprisingly quick proof of the gonality conjecture
of Green and the second author, asserting that one can read off the gonality of
a curve C from its resolution in the embedding defined by any one line bundle
of sufficiently large degree. More generally, we establish a necessary and
sufficient condition for the asymptotic vanishing of the weight one syzygies of
the module associated to an arbitrary line bundle on C.
