We have measured the spin-dependent structure function g1p of the proton in deep inelastic scattering of polarized muons off polarized protons, in the kinematic range 0.003 < x < 0.7 and 1 GeV 2 < Q2 < 60 GeV 2. Its first moment, ∫ 01g1p( x) dx, is found to be 0.136±0.011 (stat.)± 0.011 (syst.) at Q2 = 10GeV 2. This value is smaller than the prediction of the Ellis-Jaffe sum rule by two standard deviations, and is consistent with previous measurements. A combined analysis of all available proton, deuteron and neutron data confirms the Bjorken sum rule to within 10% of the theoretical value.