  We prove a homological stabilization theorem for Hurwitz spaces: moduli
spaces of branched covers of the complex projective line. This has the
following arithmetic consequence: let l>2 be prime and A a finite abelian
l-group. Then there exists Q = Q(A) such that, for q greater than Q and not
congruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t)
have the l-part of their class group isomorphic to A.
