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-bar event sample
collected with the CDF II detector. 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 results in a top
quark mass of 169.7+5.2_-4.9(stat) +/- 3.1(syst) GeV/c2.