The lifetime of the B−c meson is measured using 272 exclusive B−c→J/ψ(→μ+μ−)π− decays reconstructed in data from proton-antiproton collisions corresponding to an integrated luminosity of 6.7  fb−1 recorded by the CDF II detector at the Fermilab Tevatron. The lifetime of the B−c meson is measured to be τ(B−c)=0.452±0.048(stat)±0.027(syst)  ps. This is the first measurement of the B−c meson lifetime in a fully reconstructed hadronic channel, and it agrees with previous results and has comparable precision.