We report a measurement of the bottom-strange meson mixing phase β(s) using the time evolution of B(s)(0)→J/ψ(→μ(+)μ(-))φ(→K(+)K(-)) decays in which the quark-flavor content of the bottom-strange meson is identified at production. This measurement uses the full data set of proton-antiproton collisions at √s=1.96 TeV collected by the Collider Detector experiment at the Fermilab Tevatron, corresponding to 9.6 fb(-1) of integrated luminosity. We report confidence regions in the two-dimensional space of β(s) and the B(s)(0) decay-width difference ΔΓ(s) and measure β(s)∈[-π/2,-1.51]∪[-0.06,0.30]∪[1.26,π/2] at the 68% confidence level, in agreement with the standard model expectation. Assuming the standard model value of β(s), we also determine ΔΓ(s)=0.068±0.026(stat)±0.009(syst) ps(-1) and the mean B(s)(0) lifetime τ(s)=1.528±0.019(stat)±0.009(syst) ps, which are consistent and competitive with determinations by other experiments.