The B (s) (0) -> aEuro parts per thousand J/psi I center dot I center dot decay is observed in pp collision data corresponding to an integrated luminosity of 3 fb(-1) recorded by the LHCb detector at centre-of-mass energies of 7 TeV and 8 TeV. This is the first observation of this decay channel, with a statistical significance of 15 standard deviations. The mass of the B (s) (0) meson is measured to be 5367.08 +/- 0.38 +/- 0.15 MeV/c(2). The branching fraction ratio a"not sign(B (s) (0) -> aEuro parts per thousand J/psi I center dot I center dot)/a"not sign(B (s) (0) -> aEuro parts per thousand J/psi I center dot) is measured to be 0.0115 +/- A 0.0012 (-aEuro parts per thousand 0.0009) (+ 0.0005) . In both cases, the first uncertainty is statistical and the second is systematic. No evidence for non-resonant B (s) (0) -> aEuro parts per thousand J/psi I center dot K (+) K (-) or B (s) (0) -> aEuro parts per thousand J/psi K (+) K (-) K (+) K (-) decays is found.