We report a measurement of the lifetime of the Lambda_b baryon in decays to the Lambda_C+ pi- final state in a sample corresponding to 1.1 fb^-1 collected in p-pbar collisions at sqrt(s) = 1.96 TeV by the CDF II detector at the Tevatron collider. Using a sample of about 3000 fully reconstructed Lambda_b events we measure tau(Lambda_b) = 1.401 +- 0.046 (stat) +- 0.035 (syst) ps (corresponding to c.tau(Lambda_b) = 420.1 +- 13.7 (stat) +- 10.6 (syst) um, where c is the speed of light). The ratio o