Indirect measurement of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi mathvariant="normal">Na</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>23</mml:mn> </mml:mmultiscripts> <mml:mo>(</mml:mo> <

The cross section of the $^{23}\mathrm{Na}(p,\ensuremath{\gamma})^{24}\mathrm{Mg}$ reaction is dominated by direct capture at low energies relevant for stellar burning. Such cross sections can be constrained using spectroscopic factors (${C}^{2}S$) or asymptotic normalization coefficients from transfer reactions. In this work, the $^{23}\mathrm{Na}(^{3}\mathrm{He},d)^{24}\mathrm{Mg}$ reaction was measured at ${E}_{\text{lab}}=21$ MeV to extract spectroscopic factors for $^{24}\mathrm{Mg}$ states with excitation energies in ${E}_{x}=7--12\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$ using the Enge split-pole spectrograph at the Triangle Universities Nuclear Laboratory. A new nonresonant astrophysical $S$ factor and the direct capture reaction rate for the $^{23}\mathrm{Na}(p,\ensuremath{\gamma})$ reaction are calculated and presented based on this measurement. The new rate at $T&lt;0.04$ GK is $43%$ smaller than in previous studies. Rigorous treatments of uncertainties are presented using a Bayesian Markov Chain Monte Carlo (MCMC) method. Sources of uncertainties for computing the direct capture cross section are also discussed in detail.