We show that on any smooth compact connected manifold of dimension m >= 2 admitting a smooth non-trivial circle action S = \{ S-t\}(t is an element of R), St+1 = S-t, the set of weakly mixing C-infinity-diffeomorphisms which preserve both a smooth volume nu and a measurable Riemannian metric is dense in A(alpha) (M) = <(h o S(alpha )o h(-1 ): h is an element of Diff(infinity) (M,nu)\})over bar>(C infinity) for every Liouville number alpha. The proof is based on a quantitative version of the approximation by conjugation-method with explicitly constructed conjugation maps and partitions.