  We prove that the L^2-Betti numbers of a unimodular locally compact group G
coincide, up to a natural scaling constant, with the L^2-Betti numbers of the
countable equivalence relation induced on a cross section of any essentially
free ergodic probability measure preserving action of G. As a consequence, we
obtain that the reduced and un-reduced L^2-Betti numbers of G agree and that
the L^2-Betti numbers of a lattice Gamma in G equal those of G up to scaling by
the covolume of Gamma in G. We also deduce several vanishing results, including
the vanishing of the reduced L^2-cohomology for amenable locally compact
groups.
