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We will introduce the phase-field Navier-Stokes equations, and prove that they possess Leray global weak solutions. We will then show that for smooth initial data, solutions of the phase-field model converge weakly to solutions of the sharp-interface Navier-Stokes equations. In the convergence proof, an auxiliary PDE is introduced which couples the Navier-Stokes equations with the classical geometric problem of motion by mean curvature.