We have searched for associated production of the lightest chargino ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}}_{1}$ and next-to-lightest neutralino ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{Z}}_{2}$ of the minimal supersymmetric standard model in $p\overline{p}$ collisions at $\sqrt{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8$ TeV using the D0 detector at the Fermilab Tevatron collider. Data corresponding to an integrated luminosity of $12.5\ifmmode\pm\else\textpm\fi{}0.7{\mathrm{pb}}^{\ensuremath{-}1}$ were examined for events containing three isolated leptons. No evidence for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}}_{1}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{Z}}_{2}$ pair production was found. Limits on $\ensuremath{\sigma}({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}}_{1}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{Z}}_{2})B({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}}_{1}\ensuremath{\rightarrow}l\ensuremath{\nu}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{Z}}_{1})B({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{Z}}_{2}\ensuremath{\rightarrow}l\overline{l}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{Z}}_{1})$ are presented.