Signals consistent with the B-c(+)(2S) and B-c*(+)(2S) states are observed in proton-proton collisions at root 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 barc states are observed in the B-c(+)pi(+)pi(-) invariant mass spectrum, with the ground state B-c(+) reconstructed through its decay to J/psi pi(+). 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.