  We obtain uniform asymptotics for polynomials orthogonal on a fixed and
varying arc of the unit circle with a positive analytic weight function. We
also complete the proof of the large $s$ asymptotic expansion for the Fredholm
determinant with the kernel $\sin z/(\pi z)$ on the interval $[0,s]$, verifying
a conjecture of Dyson for the constant term in the expansion. In the Gaussian
Unitary Ensemble of random matrices, this determinant describes the probability
for an interval of length $s$ in the bulk scaling limit to be free from the
eigenvalues.
