The first observation of the B̄0→J/ψK+K- decay is presented with a data sample corresponding to an integrated luminosity of 1.0fb-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(B̄0→J/ψK+K-)=(2.53±0.31±0.19)×10-6, where the first uncertainty is statistical and the second is systematic. An amplitude analysis of the final state in the B̄0→J/ψK+K- decay is performed to separate resonant and nonresonant contributions in the K+K- spectrum. Evidence of the a0(980) resonance is reported with statistical significance of 3.9 standard deviations. The corresponding product branching fraction is measured to be B(B̄0→J/ψa0(980),a0(980)→K+K-)=(4.70±3.31±0.72)×10-7, yielding an upper limit of B(B̄0→J/ψa0(980),a0(980)→K+K-)<9.0×10-7 at 90% confidence level. No evidence of the resonant decay B̄0→J/ψϕ is found, and an upper limit on its branching fraction is set to be B(B̄0→J/ψϕ)<1.9×10-7 at 90% confidence level.