The 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.Received 18 February 2019Revised 26 March 2019DOI:https://doi.org/10.1103/PhysRevLett.122.191804Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.© 2019 CERN, for the LHCb CollaborationPhysics Subject Headings (PhySH)Research AreasBranching fractionParticle decaysParticle phenomenaParticles & Fields