Precision Measurement of the Deuteron 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 mathvariant="italic">d</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math>

We report on a high-statistics measurement of the deuteron spin structure function ${g}_{1}^{d}$ at a beam energy of 29 GeV in the kinematic range $0.029&lt;x&lt;0.8$ and $1&lt;{Q}^{2}&lt;10$ (GeV $/c$${)}^{2}$. The integral ${\ensuremath{\gamma}}_{1}^{d}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\int}{1}^{}{g}_{1}^{d}\mathrm{dx}$ evaluated at fixed ${Q}^{2}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ (GeV $/c$${)}^{2}$ gives $0.042\ifmmode\pm\else\textpm\fi{}0.003(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.004(\mathrm{syst})$. Combining this result with our earlier measurement of ${g}_{1}^{p}$, we find ${\ensuremath{\gamma}}_{1}^{p}\ensuremath{-}{\ensuremath{\gamma}}_{1}^{n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.163\ifmmode\pm\else\textpm\fi{}0.010(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.016(\mathrm{syst})$, which agrees with the prediction of the Bjorken sum rule with $O({\ensuremath{\alpha}}_{s}^{3})$ corrections, ${\ensuremath{\gamma}}_{1}^{p}\ensuremath{-}{\ensuremath{\gamma}}_{1}^{n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.171\ifmmode\pm\else\textpm\fi{}0.008$. We find the quark contribution to the proton helicity to be $\ensuremath{\Delta}q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.30\ifmmode\pm\else\textpm\fi{}0.06$.