The STAR Collaboration reports on the photoproduction of $pi^+pi^-$ 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 $pi^+pi^-$ invariant mass spectrum with a combination of $rho$ and $omega$ resonances and a direct $pi^+pi^-$ continuum. This is the first observation of the $omega$ in ultra-peripheral collisions, and the first measurement of $rho-omega$ interference at energies where photoproduction is dominated by Pomeron exchange. The $omega$ amplitude is consistent with the measured $gamma prightarrow omega p$ cross section, a classical Glauber calculation and the $omegarightarrowpi^+pi^-$ branching ratio. The $omega$ phase angle is similar to that observed at much lower energies, showing that the $rho-omega$ phase difference does not depend significantly on photon energy. The $rho^0$ differential cross section $dsigma/dt$ exhibits a clear diffraction pattern, compatible with scattering from a gold nucleus, with 2 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.
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