We report a search for the standard model (SM) Higgs boson based on data collected by the D0 experiment at the Fermilab Tevatron Collider, corresponding to an integrated luminosity of $260\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$. We study events with missing transverse energy and two acoplanar $b$ jets, which provide sensitivity to the $ZH$ production cross section in the $\ensuremath{\nu}\overline{\ensuremath{\nu}}b\overline{b}$ channel, and to $WH$ production when the lepton from the $W\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}$ decay is undetected. The data are consistent with the SM background expectation, and we set 95% C.L. upper limits on $\ensuremath{\sigma}(p\overline{p}\ensuremath{\rightarrow}ZH/WH)\ifmmode\times\else\texttimes\fi{}B(H\ensuremath{\rightarrow}b\overline{b})$ from $3.4/8.3$ to $2.5/6.3\text{ }\text{ }\mathrm{pb}$, for Higgs-boson masses between 105 and 135 GeV.