This paper continues, and in some sense completes, the study begun in [2]. We consider diffeomorphisms f of T = R/Z with irrational rotation number. A theorem of Denjoy states that if f ∈ C, it is conjugate to a rotation. The conjugating homeomorphims may be non-absolutely-continuous in which case the unique f-invariant measure on T is not equivalent to the Lebesgue measure μ. On the other hand, it is clear that f maps μ onto a measure equivalentto it, and it is this non-singular transformation that we proposie to study here. The natural classification theory for non-singular transformations on a Lebesgue space, which parallels the classification of von Neumann factors, is done modulo orbit-equivalence (also know as Dye-equivalence or weakequivalence). We list the main definitions and results concerning orbitequivalence, which we use, in section 1. The main contributors to the theory of orbit-equivalence are Hopf, Dye, Krieger, Connes and Woods, and we refer the reader to their works or to the forthcoming exposition [3] for the proofs. In section 2 we give a characterization of those transformationswhich are orbitequivalent to smooth diffeomorphisms of the circle. We show that every Cdiffeomorphism with irrational rotation number is of product type, and that every transformation of product type is orbit-equivalent to some C∞-diffeomorphism. Using the same ideas, one can easily show that any non-singular ergodic system is orbit-equivalent to some homeomorphism of T acting on the Legbesgue measure. Thus our results show the precise limitations imposed on the ergodic properties of mapping on T by smoothness conditions.