We report the first top-quark mass measurement that uses a cross-section constraint to improve the mass determination. This measurement is made with a dilepton tt event candidate sample collected with the Collider Detector II at Fermilab. From a data sample corresponding to an integrated luminosity of 1.2  fb−1, we measure a top-quark mass of 170.7+4.2−3.9(stat)±2.6(syst)±2.4(theory)  GeV/c2. The measurement without the cross-section constraint is 169.7+5.2−4.9(stat)±3.1(syst)  GeV/c2.