We describe a measurement of the top quark mass from events produced in pp collisions at a center-of-mass energy of 1.96 TeV, using the Collider Detector at Fermilab. We identify tt¯ candidates where both W bosons from the top quarks decay into leptons (eν, μν, or τν) from a data sample of 360  pb−1. The top quark mass is reconstructed in each event separately by three different methods, which draw upon simulated distributions of the neutrino pseudorapidity, tt 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 parametrized to form continuous probability density functions. A likelihood fit incorporating these parametrized 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/c2.