  We consider Brownian motion in a circular disk $\Omega$, whose boundary
$\p\Omega$ is reflecting, except for a small arc, $\p\Omega_a$, which is
absorbing. As $\epsilon=|\partial \Omega_a|/|\partial \Omega|$ decreases to
zero the mean time to absorption in $\p\Omega_a$, denoted $E\tau$, becomes
infinite. The narrow escape problem is to find an asymptotic expansion of
$E\tau$ for $\epsilon\ll1$. We find the first two terms in the expansion and an
estimate of the error. The results are extended in a straightforward manner to
planar domains and two-dimensional Riemannian manifolds that can be mapped
conformally onto the disk. Our results improve the previously derived expansion
for a general smooth domain, $E\tau =
\ds{\frac{|\Omega|}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ ($D$ is the
diffusion coefficient) in the case of a circular disk. We find that the mean
first passage time from the center of the disk is $E[\tau |
\x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}{\epsilon}} + \log 2
+\ds{{1/4}} + O(\epsilon)]$. The second term in the expansion is needed in real
life applications, such as trafficking of receptors on neuronal spines, because
$\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is
small. We also find the singular behavior of the probability flux profile into
$\p\Omega_a$ at the endpoints of $\p\Omega_a$, and find the value of the flux
near the center of the window.
