  Let S be a Dedekind scheme with field of functions K. We show that if X_K is
a smooth connected proper curve of positive genus over K, then it admits a
N\'eron model over S, i.e., a smooth separated model of finite type satisfying
the usual N\'eron mapping property. It is given by the smooth locus of the
minimal proper regular model of X_K over S, as in the case of elliptic curves.
When S is excellent, a similar result holds for connected smooth affine curves
different from the affine line, with locally finite type N\'eron models.
