We have measured the Lambda b lifetime using the exclusive decay Lambda b --> J/psi Lambda, based on 1.2 fb(-1) of data collected with the D0 detector during 2002-2006. From 171 reconstructed Lambda b decays, where the J/psi and Lambda are identified via the decays J/psi --> mu+ mu- and Lambda --> ppi, we measured the Lambda b lifetime to be tau(Lambda b)=1.218 (+0.130)/(-0.115) (stat) +/- 0.042(syst) ps. We also measured the B0 lifetime in the decay B0 --> J/psi(mu+ mu-)K(0)/(S)(pi+ pi-) to be tau(B0)=1.501 (+0.078)/(-0.074) (stat) +/- 0.050(syst) ps, yielding a lifetime ratio of tau(Lambda b)/tau(B0)=0.811 (+0.096)/(-0.087) (stat) +/- 0.034(syst).