We prove that the $\A$-genus vanishes on certain non-spin manifolds. Namely, $\A(M)$ vanishes on any oriented, compact, connected, smooth manifold $M$ with finite second homotopy group and endowed with non-trivial (isometric) smooth $S^1$ actions. This result extends that of Atiyah and Hirzebruch on spin manifolds endowed with smooth $S^1$ actions \c{AH} to manifolds which are not necessarily spin.\\ \hspace*{3mm} We prove such vanishing by means of the elliptic genus defined by Ochanine \c{Och1, Och2}, showing that it also has the special property of being ``rigid under $S^1$ actions" on these (not necessarily spin) manifolds.\\ \hspace*{3mm} We conclude with a non-trivial application of this new vanishing theorem by classifying the positive quaternion-K\"ahler 12-manifolds. Namely, we prove that every quaternion-K\"ahler 12-manifold with a complete metric of positive scalar curvature must be a symmetric space.