  Let K be the function field of a connected regular scheme S of dimension 1,
and let f : X -> Y be a finite cover of projective smooth and geometrically
connected curves over K with g(X) greater or equal to 2. Suppose that f can be
extended to a finite cover {\mathcal X} -> {\mathcal Y} of semi-stable models
over S (it is known that this is always possible up to finite separable
extension of K). We prove that there then exists a (unique) minimal such cover.
This gives a canonical way to extend X -> Y to a finite cover of semi-stable
models over S.
