The first observation of the decays $B^0_{(s)} to J/psi p overline{p}$ is reported, using proton-proton collision data corresponding to an integrated luminosity of $5.2{text{,fb}}^{-1}$, collected with the mbox{LHCb}xspace 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 begin{align*} mathcal{B}(B^0 to J/psi p overline{p}) &= (4.51pm 0.40; text{(stat)} pm 0.44; text{(syst)}) times 10^{-7}, mathcal{B}(B^0_s to J/psi p overline{p}) &= (3.58pm 0.19; text{(stat)} pm 0.39; text{(syst)}) times 10^{-6}. end{align*} For the $B^0_s$ meson, the result is much higher than the expected value of $ {cal 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 ${5279.74 pm 0.30; ({rm stat})pm 0.10; ({rm syst})}$ MeV, and the $B^0_s$ mass as ${5366.85 pm 0.19; ({rm stat})pm 0.13; ({rm syst})}$ MeV.
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