Summary: We consider the coupled elliptic system of constraints on the Cauchy data for Einstein’s equations obtained by the conformal method when the initial manifold is asymptotically Euclidean and has a non-zero mean extrinsic curvature \(\tau\). We prove that this system admits asymptotically Euclidean solutions when \(\nabla\tau\) is small in an appropriate norm and the scalar curvature of the conformal metric \(g\) is either everywhere negative with fall off conditions that we specify, or such that \(g\) belongs to the positive Yamabe class.