A measurement of the top-quark pair-production cross section in pp collisions at s√=1.96  TeV using data corresponding to an integrated luminosity of 1.12  fb−1 collected with the Collider Detector at Fermilab is presented. Decays of top-quark pairs into the final states eν+ jets and μν+ jets are selected, and the cross section and the b-jet identification efficiency are determined using a new measurement technique which requires agreement between the measured cross sections with exactly one and with multiple identified b quarks from the top-quark decays. Assuming a top-quark mass of 175  GeV/c2, a cross section of 8.5±0.6(stat)±0.7(syst)pb is measured.