A baryonic decay of the ${B}_{c}^{+}$ meson, ${B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}p\overline{p}{\ensuremath{\pi}}^{+}$, is observed for the first time, with a significance of 7.3 standard deviations, in $pp$ collision data collected with the LHCb detector and corresponding to an integrated luminosity of $3.0\text{ }\text{ }{\text{fb}}^{\ensuremath{-}1}$ taken at center-of-mass energies of 7 and 8 TeV. With the ${B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}$ decay as the normalization channel, the ratio of branching fractions is measured to be $\mathcal{B}({B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}p\overline{p}{\ensuremath{\pi}}^{+})/\mathcal{B}({B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+})=0.14{3}_{\ensuremath{-}0.034}^{+0.039}(\text{stat})\ifmmode\pm\else\textpm\fi{}0.013(\text{syst})$. The mass of the ${B}_{c}^{+}$ meson is determined as $M({B}_{c}^{+})=6274.0\ifmmode\pm\else\textpm\fi{}1.8(\text{stat})\ifmmode\pm\else\textpm\fi{}0.4(\text{syst})\text{ }\text{ }\mathrm{MeV}/{c}^{2}$, using the ${B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}p\overline{p}{\ensuremath{\pi}}^{+}$ channel.