The AdS/CFT correspondence is used to describe five-dimensional cosmology with a big crunch singularity in terms of super-Yang-Mills theory on R times S^3 deformed by a potential which is unbounded below. Classically, a Higgs field in the dual theory rolls to infinity in finite time. But since the S^3 is finite, the unstable mode spreads quantum mechanically and the singularity is resolved when self-adjoint boundary conditions are imposed at infinity. Asymptotic freedom of the coupling governing the instability gives us computational control and the quantum spreading provides a UV cutoff on particle creation. The bulk interpretation of our result is a quantum transition from a big crunch to a big bang. An intriguing consequence of the near scale-invariance of the dual theory is that a nearly scale-invariant spectrum of stress-energy perturbations is automatically generated in the boundary theory. We comment on implications for more realistic cosmologies.