We report a measurement of the single-top-quark production cross section in 2.2  fb−1 of pp collision data collected by the Collider Detector at Fermilab at s√=1.96  TeV. Candidate events are classified as signal-like by three parallel analyses which use likelihood, matrix element, and neural network discriminants. These results are combined in order to improve the sensitivity. We observe a signal consistent with the standard model prediction, but inconsistent with the background-only model by 3.7 standard deviations with a median expected sensitivity of 4.9 standard deviations. We measure a cross section of 2.2+0.7−0.6(stat+syst)  pb, extract the Cabibbo-Kobayashi-Maskawa matrix-element value |Vtb|=0.88+0.13−0.12(stat+syst)±0.07(theory), and set the limit |Vtb|>0.66 at the 95% C.L.