  A baryonic decay of the $B_c^+$ meson, $B_c^+\to J/\psi p\overline{p}\pi^+$,
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 $3.0$ fb$^{-1}$ taken at
center-of-mass energies of $7$ and $8$ $\mathrm{TeV}$. With the $B_c^+\to
J/\psi \pi^+$ decay as normalization channel, the ratio of branching fractions
is measured to be \begin{equation*} \frac{\mathcal{B}(B_c^+\to J/\psi
p\overline{p}\pi^+)}{\mathcal{B}(B_c^+\to J/\psi \pi^+)} =
0.143^{\,+\,0.039}_{\,-\,0.034}\,(\mathrm{stat})\pm0.013\,(\mathrm{syst}).
\end{equation*} The mass of the $B_c^+$ meson is determined as
$M(B_c^+)=6274.0\pm1.8\,(\mathrm{stat})\pm0.4\,(\mathrm{syst})\,\mathrm{MeV}/c^2$,
using the $B_c^+\to J/\psi p\overline{p}\pi^+$ channel.
