We present a measurement of the electron charge asymmetry in $p\overline{p}\ensuremath{\rightarrow}W+X\ensuremath{\rightarrow}e\ensuremath{\nu}+X$ events at a center-of-mass energy of 1.96 TeV, using data corresponding to $9.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected with the D0 detector at the Fermilab Tevatron Collider. The asymmetry is measured as a function of the electron pseudorapidity and is presented in five kinematic bins based on the electron transverse energy and the missing transverse energy in the event. The measured asymmetry is compared with next-to-leading-order predictions in perturbative quantum chromodynamics and provides accurate information for the determination of parton distribution functions of the proton. This is the most precise lepton charge asymmetry measurement to date.