The first observation of the B¯¯¯0→J/ψK+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¯¯¯0→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 B¯¯¯0→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(B¯¯¯0→J/ψa0(980),a0(980)→K+K−)=(4.70±3.31±0.72)×10−7, yielding an upper limit of B(B¯¯¯0→J/ψa0(980),a0(980)→K+K−)<9.0×10−7 at 90% confidence level. No evidence of the resonant decay B¯¯¯0→J/ψϕ is found, and an upper limit on its branching fraction is set to be B(B¯¯¯0→J/ψϕ)<1.9×10−7 at 90% confidence level.