The production of the rho(770)(0) meson has been measured at midrapidity (vertical bar y vertical bar <0.5) in pp and centrality differential Pb-Pb collisions at root S-NN = 2.76 TeV with the ALICE detector at the Large Hadron Collider. The particles have been reconstructed in the rho(770)(0) -> pi(+)pi(-) decay channel in the transverse-momentum (p(T)) range 0.5-11 GeV/c. A centrality-dependent suppression of the ratio of the integrated yields 2 rho(770)(0)/(pi(+)+pi(-)) is observed. The ratio decreases by similar to 40% from pp to central Pb-Pb collisions. A study of the p(T)-differential 2 rho(770)(0)/(pi(+) + pi(-)) ratio reveals that the suppression occurs at low transverse momenta, P-T <2 GeV/c. At higher momentum, particle ratios measured in heavy-ion and pp collisions are consistent. The observed suppression is very similar to that previously measured for the K*(892)(0)/K ratio and is consistent with EPOS3 predictions that may imply that rescattering in the hadronic phase is a dominant mechanism for the observed suppression.