  Green's canonical syzygy conjecture asserts a simple relationship between the
Clifford index of a smooth projective curve and the shape of the minimal free
resolution of its homogeneous ideal in the canonical embedding. We prove the
analogue of this conjecture formulated by Bayer and Eisenbud for a class of
non-reduced curves called ribbons. Our proof uses the results of Voisin and
Hirschowitz-Ramanan on Green's conjecture for general smooth curves.
