  We report evidence for $s$-channel single-top-quark production in
proton-antiproton collisions at center-of-mass energy $\sqrt{s}= 1.96
\mathrm{TeV}$ using a data set that corresponds to an integrated luminosity of
$9.4 \mathrm{fb}^{-1}$ collected by the Collider Detector at Fermilab. We
select events consistent with the $s$-channel process including two jets and
one leptonically decaying $W$ boson. The observed significance is $3.8$
standard deviations with respect to the background-only prediction. Assuming a
top-quark mass of $172.5 \mathrm{GeV}/c^2$, we measure the $s$-channel cross
section to be $1.41^{+0.44}_{-0.42} \mathrm{pb}$.
