Two-photon production of a neutral-pion pair is uniquely predicted near threshold by the theory of chiral symmetry. The prediction vanishes at the tree level and is nonzero only at one-loop order, yielding a finite result without any unknown counterterms. In this paper we calculate the cross section for both on-shell and off-shell photons, as both cases can both be studied at e+e- storage rings. This reaction is the most accessible process which directly probes the loop structure of chiral-SU(2) symmetry.