We report a measurement of the top quark mass, M_t, in the dilepton decay channel of $t\bar{t}\to b\ell'^{+}ν_{\ell'}\bar{b}\ell^{-}\bar{\nu}_{\ell}$ using an integrated luminosity of 1.0 fb^{-1} of p\bar{p} collisions collected with the CDF II detector. We apply a method that convolutes a leading-order matrix element with detector resolution functions to form event-by-event likelihoods; we have enhanced the leading-order description to describe the effects of initial-state radiation. The joint likelihood is the product of the likelihoods from 78 candidate events in this sample, which yields a measurement of M_{t} = 164.5 ± 3.9(\textrm{stat.}) ± 3.9(\textrm{syst.}) \mathrm{GeV}/c^2, the most precise measurement of M_t in the dilepton channel. ; Comment: 7 pages, 2 figures, version includes changes made prior to publication by journal