We present a measurement of the semileptonic mixing asymmetry for ${B}^{0}$ mesons, ${a}_{\mathrm{sl}}^{d}$, using two independent decay channels: ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{D}^{\ensuremath{-}}X$, with ${D}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$; and ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{D}^{*\ensuremath{-}}X$, with ${D}^{*\ensuremath{-}}\ensuremath{\rightarrow}{\overline{D}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${\overline{D}}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ (and charge conjugate processes). We use a data sample corresponding to $10.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$, collected with the D0 experiment at the Fermilab Tevatron collider. We extract the charge asymmetries in these two channels as a function of the visible proper decay length of the ${B}^{0}$ meson, correct for detector-related asymmetries using data-driven methods, and account for dilution from charge-symmetric processes using Monte Carlo simulation. The final measurement combines four signal visible proper decay length regions for each channel, yielding ${a}_{\mathrm{sl}}^{d}=[0.68\ifmmode\pm\else\textpm\fi{}0.45(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.14(\mathrm{syst})]%$. This is the single most precise measurement of this parameter, with uncertainties smaller than the current world average of $B$ factory measurements.