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(-3.9)(+4.2)(stat)+/-2.6(syst)+/-2.4(theory) GeV/c(2). The measurement without the cross-section constraint is 169.7(-4.9)(+5.2)(stat)+/-3.1(syst) GeV/c(2).