Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. 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 bc 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.