We report a measurement of the Lambda0(b) lifetime in the
exclusive decay Lambda0(b) --> J/psi Lambda0 in p anti-p collisions at s**(1/2) =
1.96 TeV using an integrated luminosity of 1.0 fb-1 of data collected by
the CDF II detector at the Fermilab Tevatron. Using fully reconstructed
decays, we measure tau(Lambda0(b)) = 1.593+0.083_-0.078 (stat.) +/- 0.033
(syst.) ps. This is the single most precise measurement of tau(Lambda0(b))
and is 3.2 sigma higher than the current world average.