We performed an improved search for nu(mu) --> nu(e) oscillation with the KEK to Kamioka (K2K) long-baseline neutrino oscillation experiment, using the full data sample of 9.2 x 10(19) protons on target. No evidence for a nu(e) appearance signal was found, and we set bounds on the nu(mu) --> nu(e) oscillation parameters. At Deltam(2)=2.8 x 10(-3) eV(2), the best-fit value of the K2Knu(mu) disappearance analysis, we set an upper limit of sin(2)2theta(mue) < 0.13 at a 90% confidence level.