  Orthogonal polynomials on the unit circle are completely determined by their
reflection coefficients through the Szeg\H{o} recurrences. We assume that the
reflection coefficients converge to some complex number a with 0 < |a| < 1. The
polynomials then live essentially on the arc {e^(i theta): alpha <= theta <= 2
pi - alpha} where cos alpha/2 = sqrt(1-|a|^2) with 0 <= alpha <= 2 pi. We
analyze the orthogonal polynomials by comparing them with the orthogonal
polynomials with constant reflection coefficients, which were studied earlier
by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under
certain assumptions on the rate of convergence of the reflection coefficients
the orthogonality measure will be absolutely continuous on the arc. In
addition, we also prove the unit circle analogue of M. G. Krein's
characterization of compactly supported nonnegative Borel measures on the real
line whose support contains one single limit point in terms of the
corresponding system of orthogonal polynomials.
