We present a measurement of the t[overline t] pair production cross section in p[overline p] collisions at sqrt(s)=1.96 TeV utilizing approximately 425 pb-1 of data collected with the D0 detector. We consider decay channels containing two high pT charged leptons (either e or µ) from leptonic decays of both top-daughter W bosons. These were gathered using four sets of selection criteria, three of which required that a pair of fully identified leptons (i.e., eµ, ee, or µµ) be found. The fourth approach imposed less restrictive criteria on one of the lepton candidates and required that at least one hadronic jet in each event be tagged as containing a b quark. For a top quark mass of 175 GeV, the measured cross section is 7.4±1.4(stat)±1.0(syst) pb and for the current Tevatron average top quark mass of 170.9 GeV, the resulting value of the cross section is 7.8±1.8(stat+syst) pb.