  We calculate the Picard group, over the integers, of the Hilbert scheme of
smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in
${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the
classes of the generators in terms of some ``natural'' divisor classes.
