The first observation of the decays B-(s)(0) -> J/psi p (p) over bar 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 beta(B-(s)(0) -> J/psi p (p) over bar). [4.51 +/- 0.40(stat)+/- 0.44(syst)] x 10(-7), BB(s)0 -> J/psi p (p) over bar) = 3.58 +/- 0.19(stat) 0.39(syst)] x 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.