  We have measured the spin-dependent structure function $g_1^p$ of the proton
in deep inelastic scattering of polarized muons off polarized protons, in the
kinematic range $0.003<x<0.7$ and $1\,\mbox{GeV}^2<Q^2<60\,\mbox{GeV}^2$. Its
first moment, $\int_0^1 g_1^p(x) dx $, is found to be $0.136 \pm
0.011\,(\mbox{stat.})\pm 0.011\,(\mbox{syst.})$ at $Q^2=10\,\mbox{GeV}^2$. 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.
