We report a measurement of the top-quark mass using events collected by the CDF II detector from pp collisions at square root of s = 1.96 TeV at the Fermilab Tevatron. We calculate a likelihood function for the top-quark mass in events that are consistent with tt --> bl(-)nu(l)bl'+ nu'(l) decays. The likelihood is formed as the convolution of the leading-order matrix element and detector resolution functions. The joint likelihood is the product of likelihoods for each of 33 events collected in 340 pb(-1) of integrated luminosity, yielding a top-quark mass M(t) = 165.2 +/- 6.1(stat) +/- 3.4(syst) GeV/c2. This first application of a matrix-element technique to tt --> bl+ nu(l)bl'- nu(l') decays gives the most precise single measurement of M(t) in dilepton events. Combined with other CDF run II measurements using dilepton events, we measure M(t) = 167.9 +/- 5.2(stat) +/- 3.7(syst) GeV/c2.