Precision Determination of the Neutron 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>n</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math>

We report on a precision measurement of the neutron spin structure function ${g}_{1}^{n}$ using deep inelastic scattering of polarized electrons by polarized ${}^{3}\mathrm{He}$. For the kinematic range $0.014&lt;x&lt;0.7$ and $1&lt;{Q}^{2}&lt;17(\mathrm{GeV}/c{)}^{2}$, we obtain $\ensuremath{\int}{0.014}^{0.7}{g}_{1}^{n}(x)dx\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.036\ifmmode\pm\else\textpm\fi{}0.004(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.005(\mathrm{syst})$ at an average ${Q}^{2}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5(\mathrm{GeV}/c{)}^{2}$. We find relatively large negative values for ${g}_{1}^{n}$ at low $x$. The results call into question the usual Regge theory method for extrapolating to $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ to find the full neutron integral $\ensuremath{\int}{1}^{}{g}_{1}^{n}(x)\mathrm{dx}$, needed for testing the quark-parton model and QCD sum rules.