Let M be an m-dimensional differentiable manifold with a nontrivial circle action S = { S t } t ∈ R , S t + 1 = S t , preserving a smooth volume μ. For any Liouville number α we construct a sequence of area-preserving diffeomorphisms H n such that the sequence H n ○ S α ○ H n −1 converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1]. For m = 2 and M equal to the unit disc D 2 = { x 2 + y 2 ⩽ 1 } or the closed annulus A = T × [ 0 , 1 ] this result proves the following dichotomy: α ∈ R ∖ Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals α (on at least one of the boundaries in the case of A ). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if α is Diophantine, then any area preserving diffeomorphism with rotation number α on the boundary (on at least one of the boundaries in the case of A ) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.