Precision Measurement of the Proton Spin Structure Function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math>

We have measured the ratio $\frac{{g}_{1}^{p}}{{F}_{1}^{p}}$ over the range $0.029&lt;x&lt;0.8$ and $1.3&lt;{Q}^{2}&lt;10$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$ using deep-inelastic scattering of polarized electrons from polarized ammonia. An evaluation of the integral $\ensuremath{\int}{0}^{1}{g}_{1}^{p}(x, {Q}^{2})\mathrm{dx}$ at fixed ${Q}^{2}=3$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$ yields 0.127\ifmmode\pm\else\textpm\fi{}0.004(stat)\ifmmode\pm\else\textpm\fi{}0.010(syst), in agreement with previous experiments, but well below the Ellis-Jaffe sum rule prediction of 0.160\ifmmode\pm\else\textpm\fi{}0.006. In the quark-parton model, this implies $\ensuremath{\Delta}q=0.27\ifmmode\pm\else\textpm\fi{}0.10$.