We present a measurement of the forward-backward charge asymmetry (${A}_{FB}$) in $p\overline{p}\ensuremath{\rightarrow}Z/{\ensuremath{\gamma}}^{*}+X\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}+X$ events at a center-of-mass energy of 1.96 TeV using $1.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected with the D0 detector at the Fermilab Tevatron collider. ${A}_{FB}$ is measured as a function of the invariant mass of the electron-positron pair, and found to be consistent with the standard model prediction. We use the ${A}_{FB}$ measurement to extract the effective weak mixing angle ${sin}^{2}{\ensuremath{\theta}}_{W}^{\mathrm{eff}}=0.2326\ifmmode\pm\else\textpm\fi{}0.0018(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.0006(\mathrm{syst})$.