An amplitude analysis of the final state structure in the B¯¯¯0s→J/ψK+K− decay mode is performed using 1.0 fb−1 of data collected by the LHCb experiment in 7 TeV center-of-mass energy pp collisions produced by the LHC. A modified Dalitz plot analysis of the final state is performed using both the invariant mass spectra and the decay angular distributions. Resonant structures are observed in the K+K− mass spectrum as well as a significant nonresonant S-wave contribution over the entire K+K− mass range. The largest resonant component is the ϕ(1020), accompanied by f0(980), f′2(1525), and four additional resonances. The overall branching fraction is measured to be B(B¯¯¯0s→J/ψK+K−)=(7.70±0.08±0.39±0.60)×10−4, where the first uncertainty is statistical, the second systematic, and the third due to the ratio of the number of B¯¯¯0s to B− mesons produced. The mass and width of the f′2(1525) are measured to be 1522.2±2.8+5.3−2.0 MeV and 84±6+10−5 MeV, respectively. The final state fractions of the other resonant states are also reported.