he first observation of the decays B0(s)→J/ψp¯p is reported, using proton-proton collision data corresponding to an integrated luminosity of 5.2  fb−1, collected with the LHCb detector. These decays are suppressed due to limited available phase space, as well as due to Okubo-Zweig-Iizuka or Cabibbo suppression. The measured branching fractions are B(B0→J/ψp¯p)=[4.51±0.40(stat)±0.44(syst)]×10−7, B(B0s→J/ψp¯p)=[3.58±0.19(stat)±0.39(syst)]×10−6. For the B0s meson, the result is much higher than the expected value of O(10−9). The small available phase space in these decays also allows for the most precise single measurement of both the B0 mass as 5279.74±0.30(stat)±0.10(syst)  MeV and the B0s mass as 5366.85±0.19(stat)±0.13(syst)  MeV.