Let R R be any ergodic, countable generic equivalence relation on a perfect Polish space X X . It follows from the main theorem of § 1 \S 1 that, modulo a meagre subset of X , R X,R may be identified with the relation of orbit equivalence ensuing from a canonical action of Z {\mathbf {Z}} . Answering a longstanding problem of Kaplansky, Takenouchi and Dyer independently gave cross-product constructions of Type III A W ∗ A{W^\ast } -factors which were not von Neumann algebras. As a specialization of a much more general result, obtained in § 3 \S 3 , we show that the Dyer factor is isomorphic to the Takenouchi factor.