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We prove that the $2^\infty$-class groups of the imaginary quadratic fieldshave the distribution predicted by the Cohen-Lenstra heuristic. Given anelliptic curve E/Q with full rational 2-torsion and no rational cyclic subgroupof order four, we analogously prove that the $2^\infty$-Selmer groups of thequadratic twists of E have distribution as predicted by Delaunay's heuristic.In particular, among the twists E^d with |d| < N, the number of curves withrank at least two is $o(N)$.

$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture