  The first search for single top quark production from the exchange of an
$s$-channel virtual $W$ boson using events with an imbalance in the total
transverse momentum, $b$-tagged jets, and no identified leptons is presented.
The full data set collected by the Collider Detector at Fermilab, corresponding
to an integrated luminosity of 9.45 fb$^{-1}$ from Fermilab Tevatron
proton-antiproton collisions at a center of mass energy of 1.96 TeV, is used.
Assuming the electroweak production of top quarks of mass 172.5 GeV/$c^2$ in
the $s$-channel, a cross section of $1.12_{-0.57}^{+0.61}$ (stat+syst) pb, with
a significance of 1.9 standard deviations, is measured. This measurement is
combined with a previous result obtained from events with an imbalance in total
transverse momentum, $b$-tagged jets, and exactly one identified lepton,
yielding a cross section of $1.36_{-0.32}^{+0.37}$ (stat+syst) pb, with a
significance of 4.2 standard deviations.
