The Collider Detector at Fermilab collected a unique sample of jets originating from bottom-quark fragmentation (b-jets) by selecting online proton-antiproton (pp¯) collisions with a vertex displaced from the pp¯ interaction point, consistent with the decay of a bottom-quark hadron. This data set, collected at a center-of-mass energy of 1.96 TeV, and corresponding to an integrated luminosity of 5.4 fb−1, is used to measure the Z-boson production cross section times branching ratio into bb¯. The number of Z→bb¯ events is determined by fitting the dijet-mass distribution, while constraining the dominant b-jet background, originating from QCD multijet events, with data. The result, σ(pp¯→Z)×B(Z→bb¯)=1.11±0.08(stat)±0.14(syst) nb, is the most precise measurement of this process, and is consistent with the standard-model prediction. The data set is also used to search for Higgs-boson production. No significant signal is expected in our data and the first upper limit on the cross section for the inclusive pp¯→H→bb¯ process at s=1.96 TeV is set, corresponding to 33 times the expected standard-model cross section, or σ=40.6 pb, at the 95% confidence level.