We study the asymptotic behaviour, as n -> infinity, 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 Szego 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.