The production of the ρ(770)0 meson has been measured at midrapidity (|y|<0.5) in pp and centrality differential Pb-Pb collisions at sNN= 2.76 TeV with the ALICE detector at the Large Hadron Collider. The particles have been reconstructed in the ρ(770)0→π+π- decay channel in the transverse-momentum (pT) range 0.5-11 GeV/c. A centrality-dependent suppression of the ratio of the integrated yields 2ρ(770)0/(π++π-) is observed. The ratio decreases by ∼40% from pp to central Pb-Pb collisions. A study of the pT-differential 2ρ(770)0/(π++π-) ratio reveals that the suppression occurs at low transverse momenta, pT<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.