An intense source of $^{37}\mathrm{Ar}$ was produced by the $(n,\ensuremath{\alpha})$ reaction on $^{40}\mathrm{Ca}$ by irradiating 330 kg of calcium oxide in the fast neutron breeder reactor at Zarechny, Russia. The $^{37}\mathrm{Ar}$ was released from the solid target by dissolution in acid, collected from this solution, purified, sealed into a small source, and brought to the Baksan Neutrino Observatory where it was used to irradiate 13 tonnes of gallium metal in the Russian-American gallium solar neutrino experiment SAGE. Ten exposures of the gallium to the source, whose initial strength was $409\ifmmode\pm\else\textpm\fi{}2$kCi, were carried out during the period April to September 2004. The $^{71}\mathrm{Ge}$ produced by the reaction $^{71}\mathrm{Ga}({\ensuremath{\nu}}_{e},{e}^{\ensuremath{-}})^{71}\mathrm{Ge}$ was extracted, purified, and counted. The measured production rate was $11.{0}_{\ensuremath{-}0.9}^{+1.0}\phantom{\rule{0.3em}{0ex}}\text{(stat)}\ifmmode\pm\else\textpm\fi{}0.6\phantom{\rule{0.3em}{0ex}}\text{(syst)}$ atoms of $^{71}\mathrm{Ge}$/d, which is $0.{79}_{\ensuremath{-}0.10}^{+0.09}$ of the theoretically calculated production rate. When all neutrino source experiments with gallium are considered together, there is an indication the theoretical cross section has been overestimated.