The first observation of the (B) over bar (0) -> J/psi K+K- decay is presented with a data sample corresponding to an integrated luminosity of 1.0 fb(-1) of pp collisions at a center-of-mass energy of 7 TeV collected with the LHCb detector. The branching fraction is measured to be B((B) over bar (0) -> J/psi K+K-) = (2.53 +/- 0.31 +/- 0.19) x 10(-6), where the first uncertainty is statistical and the second is systematic. An amplitude analysis of the final state in the (B) over bar (0) -> J/psi K+K- decay is performed to separate resonant and nonresonant contributions in the K+K- spectrum. Evidence of the a(0)(980) resonance is reported with statistical significance of 3.9 standard deviations. The corresponding product branching fraction is measured to be B((B) over bar (0) -> J/psi a(0)(980), a(0)(980) -> K+K-) = (4.70 +/- 3.31 +/- 0.72) x 10(-7), yielding an upper limit of B((B) over bar (0) -> J/psi a(0)(980), a(0)(980) -> K+K-) < 9.0 x 10(-7) at 90% confidence level. No evidence of the resonant decay <(B)over bar>(0) J/psi phi is found, and an upper limit on its branching fraction is set to be B((B) over bar (0) -> J/psi phi) <1.9 x 10(-7) at 90% confidence level.