We present a measurement of the B+_c meson lifetime in the
semileptonic decay mode B+_c --> J/psi e+ nu_e using the CDF II detector at
the Fermilab Tevatron Collider. From a sample of about 360 pb-1 of p anti-p
collisions at s**(1/2) = 1.96 TeV, we reconstruct J/psi e+ pairs with
invariant mass in the kinematically allowed range 4 < M_J/psi e < 6 GeV/c2.
A fit to the decay-length distribution of 238 signal events yields a
measured B+_c meson lifetime of 0.463+0.073_-0.065 (stat) +/- 0.036 (syst)
ps.