  The first observation of the decay of a $B^0$ meson to a purely baryonic
final state, $B^0 \to p \bar{p}$, is reported. The proton-proton collision data
sample used was collected with the LHCb experiment at center-of-mass energies
of 7 and 8 TeV and corresponds to an integrated luminosity of 3.0 fb$^{-1}$.
The branching fraction is determined to be $${\cal B}(B^0 \to p \bar{p}) = (
1.25 \pm 0.27 \pm 0.18 ) \times 10^{-8} ,$$ where the first uncertainty is
statistical and the second systematic. The decay mode $B^0 \to p \bar{p}$ is
the rarest decay of the $B^0$ meson observed to date. The decay $B_s^0 \to p
\bar{p}$ is also investigated. No signal is seen and the upper limit ${\cal
B}(B_s^0 \to p \bar{p}) < 1.5 \times 10^{-8}$ at 90% confidence level is set on
the branching fraction.
