The first observation of the decay of a B0 meson to a purely baryonic final state, B-0 -> p$(p)over-bar-$ , 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 B(B-0 -> p$(p)over-bar-$) = (1.25 +/- 0.27 +/- 0.18) x 10(-8), where the first uncertainty is statistical and the second systematic. The decay mode B-0 -> p$(p)over-bar-$ is the rarest decay of the B-0 meson observed to date. The decay B-s(0 )-> p$(p)over-bar-$ is also investigated. No signal is seen and the upper limit B(B-s(0) -> p$(p)over-bar-$) < 1.5 x 10(-8) at 90% confidence level is set on the branching fraction.