Abstract: We report the observation of electroweak single top quark production in 3: 2 fb(-1) of p (p) over bar collision data collected by the Collider Detector at Fermilab at root s = 1.96 TeV. Candidate events in the W + jets topology with a leptonically decaying W boson are classified as signal-like by four parallel analyses based on likelihood functions, matrix elements, neural networks, and boosted decision trees. These results are combined using a super discriminant analysis based on genetically evolved neural networks in order to improve the sensitivity. This combined result is further combined with that of a search for a single top quark signal in an orthogonal sample of events with missing transverse energy plus jets and no charged lepton. We observe a signal consistent with the standard model prediction but inconsistent with the background-only model by 5.0 standard deviations, with a median expected sensitivity in excess of 5.9 standard deviations. We measure a production cross section of 2.3-(+0.6)(0.5) (stat + sys) pb, extract the value of the Cabibbo-Kobayashi-Maskawa matrix element vertical bar V-tb vertical bar = 0.91(-0.11)(+0.11) (stat + sys) +/- 0.07 (theory), and set a lower limit vertical bar V-tb vertical bar > 0.71 at the 95% C. L., assuming m(t) = 175 GeV/c(2).