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As defined by Thompson, a Heegaard splitting H has the disjoint curve property if there is a triple of essential curves a, b, c in H where, first, a is disjoint from b and b is disjoint from c and, second, a and c bound disks on opposite sides of H. We'll prove that if H has sufficiently high genus (with respect to the ambient three-manifold) then H has the disjoint curve property. The proof is technical and relies extensively on normal surface theory.