We describe a measurement of the top quark mass using events with two charged leptons collected by the CDF II detector from pp collisions with s√=1.96  TeV at the Fermilab Tevatron. The likelihood in top quark mass is calculated for each event by convoluting the leading order matrix element describing qq→tt→bℓνℓbℓ′νℓ′ with detector resolution functions. The presence of background events in the data sample is modeled using similar calculations involving the matrix elements for major background processes. In a data sample with integrated luminosity of 340  pb−1, we observe 33 candidate events and measure Mtop=165.2±6.1(stat.)±3.4(syst.)  GeV/c2. This measurement represents the first application of this method to events with two charged leptons and is the most precise single measurement of the top quark mass in this channel.