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We present measurements of the differential cross section and Lambda recoil polarization for the gamma p -> K+ Lambda reaction made using the CLAS detector at Jefferson Lab. These measurements cover the center-of-mass energy range from 1.62 to 2.84 GeV and a wide range of center-of-mass K+ production angles. Independent analyses were performed using the K+ p pi(-) and K+ p (missing pi(-)) final-state topologies; results from these analyses were found to exhibit good agreement. These differential-cross-section measurements show excellent agreement with previous CLAS and LEPS results and offer increased precision and a 300-MeV increase in energy coverage. The recoil polarization data agree well with previous results and offer a large increase in precision and a 500-MeV extension in energy range. The increased center-of-mass energy range that these data represent will allow for independent study of nonresonant K+ Lambda photoproduction mechanisms at all production angles.