  We study the asymptotic behavior, as $n\to\infty$, of ratios of Toeplitz
determinants $D_n(e^h d\mu)/D_n(d\mu)$ defined by a measure $\mu$ on the unit
circle and a sufficiently smooth function $h$. The approach we follow is based
on the theory of orthogonal polynomials. We prove that the second order
asymptotics depends on $h$ and only a few Verblunsky coefficients associated to
$\mu$. As a result, we establish a relative version of the Strong Szeg\H{o}
Limit Theorem for a wide class of measures $\mu$ with essential support on a
single arc. In particular, this allows the measure to have a singular component
within or outside of the arc.
