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We report the first measurement at the LHC of coherent photoproduction of ρ0 mesons in ultra-peripheral Pb-Pb collisions. The invariant mass and transverse momentum distributions for ρ0 production are studied in the π+π− decay channel at mid-rapidity. The production cross section in the rapidity range |y|<0.5 is found to be dσ/dy=425±10(stat.) +42−50(sys.) mb. Coherent ρ0 production is studied with and without requirement of nuclear breakup, and the fractional yields for various breakup scenarios are presented. The results are compared with those from lower energies and with model predictions based on the Glauber model and the color dipole model. The measured cross section is found to be inconsistent with a scaling of the γ-nucleon cross section using the Glauber model.