Using pp collision data corresponding to an integrated luminosity of 8.5  fb−1 recorded by the LHCb experiment at center-of-mass energies of s=7, 8, and 13 TeV, the observation of an excited Bc+ state in the Bc+π+π− invariant-mass spectrum is reported. The observed peak has a mass of 6841.2±0.6(stat)±0.1(syst)±0.8(Bc+)  MeV/c2, where the last uncertainty is due to the limited knowledge of the Bc+ mass. It is consistent with expectations of the Bc*(2S31)+ state reconstructed without the low-energy photon from the Bc*(1S31)+→Bc+γ decay following Bc*(2S31)+→Bc*(1S31)+π+π−. A second state is seen with a global (local) statistical significance of 2.2σ (3.2σ) and a mass of 6872.1±1.3(stat)±0.1(syst)±0.8(Bc+)  MeV/c2, and is consistent with the Bc(2S10)+ state. These mass measurements are the most precise to date.