52 pages, 23 figures, 5 tables. Submitted to Physical Review DWe describe a measurement of the top quark mass from events produced in proton-antiproton collisions at a center-of-mass energy of 1.96 TeV, using the Collider Detector at Fermilab. We identify top-antitop candidates where both W bosons from the top quarks decay into leptons (electron, muon, or tau plus neutrino) from a data sample of 360 inverse picobarns. The top quark mass is reconstructed in each event separately by three different methods, which draw upon simulated distributions of the neutrino pseudorapidity, top-antitop longitudinal momentum, or neutrino azimuthal angle in order to extract probability distributions for the top quark mass. For each method, representative mass distributions, or templates, are constructed from simulated samples of signal and background events, and parameterized to form continuous probability density functions. A likelihood fit incorporating these parameterized templates is then performed on the data sample masses in order to derive a final top quark mass. Combining the three template methods, taking into account correlations in their statistical and systematic uncertainties, results in a top quark mass measurement of 170.1 +/- 6.0 (stat) +/- 4.1 (syst) GeV/c^2
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