  We prove that ribbons, i.e. double structures associated with a line bundle
$\SE$ over its reduced support, a smooth irreducible projective curve of
arbitrary genus, are smoothable if their arithmetic genus is greater than or
equal to $3 $ and the support curve possesses a smooth irreducible double cover
with trace zero module $\SE$. The method we use is based on the infinitesimal
techniques that we develop to show that if the support curve admits such a
double cover then every embedded ribbon over the curve is ``infinitesimally
smoothable'', i.e. the ribbon can be obtained as central fiber of the image of
some first--order infinitesimal deformation of the map obtained by composing
the double cover with the embedding of the reduced support in the ambient
projective space containing the ribbon. We also obtain embeddings in the same
projective space for all ribbons associated with $\SE$. Then, assuming the
existence of the double cover, we prove that the ``infinitesimal smoothing''
can be extended to a global embedded smoothing for embedded ribbons of
arithmetic genus greater than or equal to 3. As a consequence we obtain the
smoothing results.
