  In this paper we prove that, under an explicit integral pinching assumption
between the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar
curvature, a closed 3-manifold with positive scalar curvature admits an
Einstein metric with positive curvature. In particular this implies that the
manifold is diffeomorphic to a quotient of ${\Bbb S}^3$.
