We present results from a search for $WZ$ production with subsequent decay to $\ensuremath{\ell}\ensuremath{\nu}{\ensuremath{\ell}}^{\ensuremath{'}}{\overline{\ensuremath{\ell}}}^{\ensuremath{'}}$ ($\ensuremath{\ell}$ and ${\ensuremath{\ell}}^{\ensuremath{'}}=e$ or $\ensuremath{\mu}$) using $0.30\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected by the D0 experiment between 2002 and 2004 at the Fermilab Tevatron. Three events with $WZ$ decay characteristics are observed. With an estimated background of $0.71\ifmmode\pm\else\textpm\fi{}0.08$ events, we measure the $WZ$ production cross section to be ${4.5}_{\ensuremath{-}2.6}^{+3.8}\text{ }\text{ }\mathrm{pb}$, with a 95% C.L. upper limit of 13.3 pb. The 95% C.L. limits for anomalous $WWZ$ couplings are found to be $\ensuremath{-}2.0<\ensuremath{\Delta}{\ensuremath{\kappa}}_{Z}<2.4$ for form factor scale $\ensuremath{\Lambda}=1\text{ }\text{ }\mathrm{TeV}$, and $\ensuremath{-}0.48<{\ensuremath{\lambda}}_{Z}<0.48$ and $\ensuremath{-}0.49<\ensuremath{\Delta}{g}_{1}^{Z}<0.66$ for $\ensuremath{\Lambda}=1.5\text{ }\text{ }\mathrm{TeV}$.