Signals consistent with the B_{c}^{+}(2S) and B_{c}^{*+}(2S) states are observed in proton-proton collisions at sqrt[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[over ¯]c states are observed in the B_{c}^{+}π^{+}π^{-} invariant mass spectrum, with the ground state B_{c}^{+} 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 B_{c}^{+}(2S) meson is measured to be 6871.0±1.2(stat)±0.8(syst)±0.8(B_{c}^{+}) MeV, where the last term corresponds to the uncertainty in the world-average B_{c}^{+} mass.