We present measurements of the lifetime in the exclusive decay channel J/0, with J/µ+µ- and 0p-, the B0 lifetime in the decay B0J/K with J/µ+µ- and K+-, and the ratio of these lifetimes. The analysis is based on approximately 250 pb-1 of data recorded with the D0 detector in p collisions at =1.96 TeV. The lifetime is determined to be ()=1.22(stat)±0.04(syst) ps, the B0 lifetime (B0)=1.40(stat)±0.03(syst) ps, and the ratio ()/(B0)=0.87(stat)±0.03(syst). In contrast with previous measurements using semileptonic decays, this is the first determination of the lifetime based on a fully reconstructed decay channel.