  We performed an improved search for $\nu_\mu \to \nu_e$ oscillation with the
KEK to Kamioka (K2K) long-baseline neutrino oscillation experiment, using the
full data sample of $9.2 \times 10^{19}$\xspace protons on target. No evidence
for a $\nu_e$ appearance signal was found, and we set bounds on the $\nu_\mu
\to \nu_e$ oscillation parameters. At $\Delta m^2$ = $2.8 \times 10^{-3}
\mathrm{eV}^2$, the best fit value of the K2K $\nu_\mu$ disappearance analysis,
we set an upper limit of $\sin^2 2\theta_{\mu e}$ $<$ 0.13 at 90% confidence
level.
