  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 $t\bar{t}$ 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 $\rm{170.7^{+4.2}_{-3.9}(stat)\pm2.6(syst)}$
$\rm{\pm2.4(theory) GeV/{\it{c}}^{2}}$. The measurement without the cross
section constraint results in a top quark mass of
$\rm{169.7^{+5.2}_{-4.9}(stat)\pm3.1(syst) GeV/{\it{c}}^{2}}$.
