The results of a search for the rare two-body charmless baryonic decays $B^0\to p \bar{p}$ and $B_s^0 \to p \bar{p}$ are reported. The analysis uses a datasample, corresponding to an integrated luminosity of 0.9 fb$^{-1}$, of $pp$collision data collected by the LHCb experiment at a centre-of-mass energy of 7TeV. An excess of $B^0 \to p \bar{p}$ candidates with respect to backgroundexpectations is seen with a statistical significance of 3.3 standarddeviations. This is the first evidence for a two-body charmless baryonic $B^0$decay. No significant $B_s^0 \to p \bar{p}$ signal is observed, leading to animprovement of three orders of magnitude over previous bounds. If the excessevents are interpreted as signal, the 68.3% confidence level intervals on thebranching fractions are {eqnarray} \cal{B}(B^0 \to p \bar{p}) & = & (1.47\,^{+0.62}_{-0.51} \,^{+0.35}_{-0.14}) \times 10^{-8} \,, *{0.3cm}\cal{B}(B_s^0 \to p \bar{p}) & = & (2.84 \,^{+2.03}_{-1.68} \,^{+0.85}_{-0.18})\times 10^{-8} \,, {eqnarray} where the first uncertainty is statistical andthe second is systematic.