The first observation of the decays B_{(s)}^{0}→J/ψpp[over ¯] 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(B^{0}→J/ψpp[over ¯])=[4.51±0.40(stat)±0.44(syst)]×10^{-7}, B(B_{s}^{0}→J/ψpp[over ¯])=[3.58±0.19(stat)±0.39(syst)]×10^{-6}. For the B_{s}^{0} 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 B^{0} mass as 5279.74±0.30(stat)±0.10(syst) MeV and the B_{s}^{0} mass as 5366.85±0.19(stat)±0.13(syst) MeV.