We present a measurement of the top-quark pair-production cross section in pp collisions at s√=1.96  TeV using a data sample corresponding to 1.7  fb−1 of integrated luminosity collected with the Collider Detector at Fermilab. We reconstruct tt events in the lepton+jets channel, consisting of eν+jets and μν+jets final states. The dominant background is the production of W bosons in association with multiple jets. To suppress this background, we identify electrons from the semileptonic decay of heavy-flavor jets (“soft electron tags”). From a sample of 2196 candidate events, we obtain 120 tagged events with a background expectation of 51±3 events, corresponding to a cross section of σtt=7.8±2.4(stat)±1.6(syst)±0.5(lumi)  pb. We assume a top-quark mass of 175  GeV/c2. This is the first measurement of the tt cross section with soft electron tags in run II of the Tevatron.