We describe a search for $Z$ boson pair production in $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ with the D0 detector at the Fermilab Tevatron Collider using a data sample corresponding to an integrated luminosity of $2.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Using the final state decay $ZZ\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ (where $\ensuremath{\ell}=e$ or $\ensuremath{\mu}$) we find a signal with a 2.6 standard deviations significance (2.0 expected) corresponding to a cross section of $\ensuremath{\sigma}(p\overline{p}\ensuremath{\rightarrow}ZZ+X)=2.01\ifmmode\pm\else\textpm\fi{}0.93(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.29(\mathrm{sys})\text{ }\text{ }\mathrm{pb}$.