© 2017 American Physical Society. The STAR Collaboration reports on the photoproduction of π+π- pairs in gold-gold collisions at a center-of-mass energy of 200 GeV/nucleon-pair. These pion pairs are produced when a nearly real photon emitted by one ion scatters from the other ion. We fit the π+π- invariant-mass spectrum with a combination of ρ0 and ω resonances and a direct π+π- continuum. This is the first observation of the ω in ultraperipheral collisions, and the first measurement of ρ-ω interference at energies where photoproduction is dominated by Pomeron exchange. The ω amplitude is consistent with the measured γp→ωp cross section, a classical Glauber calculation, and the ω→π+π- branching ratio. The ω phase angle is similar to that observed at much lower energies, showing that the ρ-ω phase difference does not depend significantly on photon energy. The ρ0 differential cross section dσ/dt exhibits a clear diffraction pattern, compatible with scattering from a gold nucleus, with two minima visible. The positions of the diffractive minima agree better with the predictions of a quantum Glauber calculation that does not include nuclear shadowing than with a calculation that does include shadowing.