An updated measurement of the single top quark production cross section is presented using the full data set collected by the Collider Detector at Fermilab (CDF) and corresponding to 9.5 fb${}^{-1}$ of integrated luminosity from proton-antiproton collisions at 1.96 TeV center-of-mass energy. The events selected contain an imbalance in the total transverse energy, jets identified as originating from $b$ quarks, and no identified leptons. The sum of the $s$- and $t$-channel single top quark cross sections is measured to be $3.53_{-1.16}^{+1.25}$ pb and a lower limit on $V_{tb}$ of 0.63 is obtained at the 95% credibility level. These measurements are combined with previously reported CDF results obtained from events with an imbalance in total transverse energy, jets identified as originating from $b$ quarks, and exactly one identified lepton. The combined cross section is measured to be $3.02_{-0.48}^{+0.49}$ pb and a lower limit on $V{tb}$ of 0.84 is obtained at the 95% credibility level.
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