We measure the dimuon charge asymmetry $A$ in $p\overline{p}$ collisions at a center of mass energy $\sqrt{s}=1960\text{ }\text{ }\mathrm{GeV}$. The data was recorded with the D0 detector and corresponds to an integrated luminosity of approximately $1.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Assuming that the asymmetry $A$ is due to asymmetric ${B}^{0}\ensuremath{\leftrightarrow}{\overline{B}}^{0}$ mixing and decay, we extract the $CP$-violation parameter of ${B}^{0}$ mixing and decay: $\frac{\ensuremath{\mathfrak{R}}({ϵ}_{{B}^{0}})}{1+|{ϵ}_{{B}^{0}}{|}^{2}}=\frac{{A}_{{B}^{0}}}{4}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.0023\ifmmode\pm\else\textpm\fi{}0.0011(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.0008(\mathrm{syst}).$${A}_{{B}^{0}}$ is the dimuon charge asymmetry from decays of ${B}^{0}{\overline{B}}^{0}$ pairs. The general case, with $CP$ violation in both ${B}^{0}$ and ${B}_{s}^{0}$ systems, is also considered. Finally we obtain the forward-backward asymmetry that quantifies the tendency of ${\ensuremath{\mu}}^{+}$ to go in the proton direction and ${\ensuremath{\mu}}^{\ensuremath{-}}$ to go in the antiproton direction. The results are consistent with the standard model and constrain new physics.