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The STAR Collaboration reports the first observation of exclusive rho(0) photoproduction, AuAu-->AuAurho(0), and rho(0) production accompanied by mutual nuclear Coulomb excitation, AuAu-->Au*Au*rho(0), in ultraperipheral heavy-ion collisions. The rho(0) have low transverse momenta, consistent with coherent coupling to both nuclei. The cross sections at sqrt[s(NN)]=130 GeV agree with theoretical predictions treating rho(0) production and Coulomb excitation as independent processes.