Using the exclusive decay B0s-->J/psi(mu+mu-)phi(K+K-), we report the most precise single measurement of the B0s lifetime. The data sample corresponds to an integrated luminosity of approximately 220 pb(-1) collected with the D0 detector at the Fermilab Tevatron Collider in 2002-2004. We reconstruct 337 signal candidates, from which we extract the B0s lifetime, tau(B0s)=1.444(+0.098)(-0.090)(stat)+/-0.020(sys) ps. We also report a measurement for the lifetime of the B0 meson using the exclusive decay B0-->J/psi(mu+mu-)K*0(892)(K+pi-). We reconstruct 1370 signal candidates, obtaining tau(B0)=1.473(+0.052)(-0.050)(stat)+/-0.023(sys) ps, and the ratio of lifetimes, tau(B0s)/tau(B0)=0.980(+0.076)(-0.071)(stat)+/-0.003(sys).