  We construct geometric generators of the effective $S^1$-equivariant Spin-
(and oriented) bordism groups with two inverted. We apply this construction to
the question of which $S^1$-manifolds admit invariant metrics of positive
scalar curvature.
  It turns out that, up to taking connected sums with several copies of the
same manifold, the only obstruction to the existence of such a metric is an
$\hat{A}$-genus of orbit spaces. This $\hat{A}$-genus generalizes a previous
definition of Lott for orbit spaces of semi-free $S^1$-actions.
  As a further application of our results, we give a new proof of the vanishing
of the $\hat{A}$-genus of a Spin manifold with non-trivial $S^1$-action
originally proven by Atiyah and Hirzebruch. Moreover, based on our computations
we can give a bordism-theoretic proof for the rigidity of elliptic genera
originally proven by Taubes and Bott--Taubes.
