Signals consistent with the Bc+(2S) and Bc*+(2S) states are observed in proton-proton collisions at s=13  TeV, in an event sample corresponding to an integrated luminosity of 143  fb-1, collected by the CMS experiment during the 2015–2018 LHC running periods. These excited b¯c states are observed in the Bc+π+π- invariant mass spectrum, with the ground state Bc+ reconstructed through its decay to J/ψπ+. The two states are reconstructed as two well-resolved peaks, separated in mass by 29.1±1.5(stat)±0.7(syst)  MeV. The observation of two peaks, rather than one, is established with a significance exceeding five standard deviations. The mass of the Bc+(2S) meson is measured to be 6871.0±1.2(stat)±0.8(syst)±0.8(Bc+)  MeV, where the last term corresponds to the uncertainty in the world-average Bc+ mass.