This report describes a search for associated production of $W$ and Higgs bosons based on data corresponding to an integrated luminosity of $\mathcal{L}\ensuremath{\approx}5.3\text{ }\text{ }\mathrm{f}{\mathrm{b}}^{\ensuremath{-}1}$ collected with the D0 detector at the Fermilab Tevatron $p\overline{p}$ Collider. Events containing a $W\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}$ candidate (with $\ensuremath{\ell}$ corresponding to $e$ or $\ensuremath{\mu}$) are selected in association with two or three reconstructed jets. One or two of the jets are required to be consistent with having evolved from a $b$ quark. A multivariate discriminant technique is used to improve the separation of signal and backgrounds. Expected and observed upper limits are obtained for the product of the $WH$ production cross section and branching ratios and reported in terms of ratios relative to the prediction of the standard model as a function of the mass of the Higgs boson (${M}_{H}$). The observed and expected 95% C.L. upper limits obtained for an assumed ${M}_{H}=115\text{ }\text{ }\mathrm{GeV}$ are, respectively, factors of 4.5 and 4.8 larger than the value predicted by the standard model.