In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {αn}∞n=0be a sequence in the open unit disk in the complex plane, letB0=1andBn(z)=∏k=0nαk|αk|αk−z1−αkz,n=1,2,…,(αk/|αk|=−1 whenαk=0), and letL=span{B:n=0,1,2,…}.We consider the following “moment” problem: Given a positive-definite Hermitian inner product ⦠·,·⦔ on L×L, find a non-decreasing functionμon [−π,π] (or a positive Borel measureμon [−π,π)) such that⦠f,g⦔=∫π−πf(eiθ)g(eiθ)dμ(θ) forf,g∈L.In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that∑n=1∞(1−|αn|)<∞.If this series diverges the solution is always unique.