Inelastic magnetic neutron scattering reveals a localized spin resonance at 4.5 meV in the ordered phase of the geometrically frustrated cubic antiferromagnet ZnCr2O4. The resonance develops abruptly from quantum critical fluctuations upon cooling through a first order transition to a co-planar antiferromagnet at T(c) = 12. 5(5) K. We argue that this transition is a three dimensional analog of the spin-Peierls transition.