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 $\ell > 2$ be prime and $A$ a finite abelian $\ell$-group. Then there exists $Q = Q(A)$ such that, for $q$ greater than $Q$, a positive fraction of quadratic extensions of $\mathbb{F}_q(t)$ have the $\ell$-part of their class group isomorphic to $A$.
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