On omega context free languages which are Borel sets of infinite rank

This paper is a continuation of the study of topological properties of omega context free languages (ω-CFL). We proved in [Topological Properties of Omega Context Free Languages, Theoretical Computer Science, Volume 262 (1-2), 2001, p. 669697] that the class of ω-CFL exhausts the finite ranks of the Borel hierarchy, and in [Borel Hierarchy and Omega Context Free Languages, Theoretical Computer Science, to appear] that there exist some ω-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical specifications of infinite computations, In:”A Decade of Concurrency” (J. W. de Bakker et al., eds), Springer LNCS 803 (1994), 583-621].