Let σ be a probability Borel measure on the unit circle T and {ϕn} be the orthonormal polynomials with respect to σ. We say that σ is a Szegő measure, if it has an arbitrary singular part σs, and ∫Tlogσ′dm>−∞, where σ′ is the density of the absolutely continuous part of σ, m being the normalized Lebesgue measure on T. The entropy integrals for ϕn are defined as ϵn=∫T|ϕn|2log|ϕn|dσ. It is not difficult to show that ϵn=o¯(n). In this paper, we construct a measure from the Szegő class for which this estimate is sharp (over a subsequence of n’s).