This Letter reports on a search for nu(mu) --> nu(e) transitions by the MINOS experiment based on a 3.14x10(20) protons-on-target exposure in the Fermilab NuMI beam. We observe 35 events in the Far Detector with a background of 27+/-5(stat)+/-2(syst) events predicted by the measurements in the Near Detector. If interpreted in terms of nu(mu) --> nu(e) oscillations, this 1.5sigma excess of events is consistent with sin2(2theta(13)) comparable to the CHOOZ limit when |Delta m2|=2.43x10(-3) eV2 and sin2(2theta(23))=1.0 are assumed.