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We present a measurement of the cross section and the first measurement of the heavy flavor content of associated direct photon + muon events produced in hadronic collisions. These measurements come from a sample of 1.8 TeV pp̅ collisions recorded with the Collider Detector at Fermilab. Quantum chromodynamics (QCD) predicts that these events are primarily due to the Compton scattering process cg→cγ , with the final-state charm quark producing a muon. The cross section for events with a photon transverse momentum between 12 and 40 GeV/c is measured to be 46.8±6.3±7.5 pb, which is two standard deviations below the most recent theoretical calculation. A significant fraction of the events in the sample contain a final-state bottom quark. The ratio of charm to bottom production is measured to be 2.4±1.2, in good agreement with QCD models.