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 sqrt[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(-0.6)(+0.7)(stat+syst) pb, extract the Cabibbo-Kobayashi-Maskawa matrix-element value |V(tb)|=0.88(-0.12)(+0.13)(stat+syst)+/-0.07(theory), and set the limit |V(tb)|>0.66 at the 95% C.L.