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We present first evidence for the production of single top quarks in the D0 detector at the Fermilab Tevatron pp collider. The standard model predicts that the electroweak interaction can produce a top quark together with an anti-bottom quark or light quark, without the antiparticle top-quark partner that is always produced from strong-coupling processes. Top quarks were first observed in pair production in 1995, and since then, single top-quark production has been searched for in ever larger data sets. In this analysis, we select events from a 0.9 fb-1 data set that have an electron or muon and missing transverse energy from the decay of a W boson from the top-quark decay, and two, three, or four jets, with one or two of the jets identified as originating from a b hadron decay. The selected events are mostly backgrounds such as W + jets and tt events, which we separate from the expected signals using three multivariate analysis techniques: boosted decision trees, Bayesian neural networks, and matrix-element calculations. A binned likelihood fit of the signal cross section plus background to the data from the combination of the results from the three analysis methods gives a cross section for single top-quark production of σ(pp Y→ tb + X, tqb + X) = 4.7 ± 1:3 pb. The probability to measure a cross section at this value or higher in the absence of signal is 0.014%, corresponding to a 3.6 standard deviation significance. The measured cross section value is compatible at the 10% level with the standard model prediction for electroweak top-quark production. We use the cross section measurement to directly determine the Cabibbo-Kobayashi- Maskawa quark mixing matrix element that describes the Wtb coupling and find |VtbƒL1 | = 1.31-0.21+0.25 , where ƒL1 is a generic vector coupling. This model-independent measurement translates into 0.68 < |Vtb| ≤ 1 at the 95% C.L. in the standard model.