we consider the conformal decomposition of einstein's constraint equations introduced by lichnerowicz and york, on a compact manifold with boundary. we show that there exists a solution to the coupled hamiltonian and momentum constraint equations when the derivative of the mean extrinsic curvature is small enough, and assuming that the ricci scalar of the background metric is bounded, though it can change sign on the manifold. the solutions are in general not uniquely determined by the source functions and boundary data. the proof technique is based on finding barriers for the hamiltonian constraint equation which are independent of the solutions of the momentum constraint equation, and then using standard fixed-point methods for increasing operators in banach spaces. this work generalizes a previous work by isenberg and moncrief on closed manifolds.