The lifetime of the B0s meson is measured using the exclusive decay mode B0s→J/ψφ, where J/ψ→μ+μ− and φ→K+K−. The data sample consists of 110pb−1 of pp ̄ collisions at √s=1.8TeV, collected by the CDF detector at the Fermilab Tevatron collider during 1992–1995. We find 58±12 signal events, and the B0s meson lifetime is determined to be τB0s=1.34+0.23−0.19(stat)±0.05(syst) ps. This result is consistent with previous measurements of the B0s, B+, and B0d meson lifetimes and with theoretical predictions.