Using $pp$ collision data corresponding to an integrated luminosity of $8.5\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ 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 ${B}_{c}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ invariant-mass spectrum is reported. The observed peak has a mass of $6841.2\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.1(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.8({B}_{c}^{+})\text{ }\text{ }\mathrm{MeV}/{c}^{2}$, 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 ${B}_{c}^{*}(1{^{3}S}_{1}{)}^{+}\ensuremath{\rightarrow}{B}_{c}^{+}\ensuremath{\gamma}$ decay following ${B}_{c}^{*}(2{^{3}S}_{1}{)}^{+}\ensuremath{\rightarrow}{B}_{c}^{*}(1{^{3}S}_{1}{)}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. A second state is seen with a global (local) statistical significance of $2.2\ensuremath{\sigma}$ ($3.2\ensuremath{\sigma}$) and a mass of $6872.1\ifmmode\pm\else\textpm\fi{}1.3(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.1(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.8({B}_{c}^{+})\text{ }\text{ }\mathrm{MeV}/{c}^{2}$, and is consistent with the ${B}_{c}(2{^{1}S}_{0}{)}^{+}$ state. These mass measurements are the most precise to date.