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 sqrt[s]=7, 8, and 13 TeV, the observation of an excited B_{c}^{+} state in the B_{c}^{+}π^{+}π^{-} invariant-mass spectrum is reported. The observed peak has a mass of 6841.2±0.6(stat)±0.1(syst)±0.8(B_{c}^{+}) MeV/c^{2}, where the last uncertainty is due to the limited knowledge of the B_{c}^{+} mass. It is consistent with expectations of the B_{c}^{*}(2^{3}S_{1})^{+} state reconstructed without the low-energy photon from the B_{c}^{*}(1^{3}S_{1})^{+}→B_{c}^{+}γ decay following B_{c}^{*}(2^{3}S_{1})^{+}→B_{c}^{*}(1^{3}S_{1})^{+}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(B_{c}^{+}) MeV/c^{2}, and is consistent with the B_{c}(2^{1}S_{0})^{+} state. These mass measurements are the most precise to date.