Using approximately 4 million hadronic Z decays recorded with the aleph detector from 1991 through 1995, the lifetime of the b baryon is measured with three independent methods. From the impact parameter distribution of candidate leptons in 1063 events with Λℓ− combinations, the average b baryon lifetime is measured to be 1.20±0.08±0.06 ps. From a sample of 193 fully reconstructed Λ+c candidates correlated with a lepton and a sample of 46 Λℓ+ℓ− combinations, the Λb lifetime is measured to be 1.21±0.11 ps. The product branching fractions to these final states are Br(b→Λb⋅BR(Λb→Λℓ−ν¯X)=0.326±0.016±0.039 % for the first sample and Br(b→Λb⋅BR(Λb→Λ+cℓ−ν¯X)=0.86±0.07±0.14 % for the second and third samples combined.