The cross section of atomic electron Compton scattering $gamma + e rightarrow gamma^prime + e^prime $ was measured in the 4.40--5.475 GeV photon beam energy region by the {em PrimEx} collaboration at Jefferson Lab with an accuracy of 2% and less. The results are consistent with theoretical predictions that include next-to-leading order radiative corrections. The measurements provide the first high precision test of this elementary QED process at beam energies greater than 0.1 GeV.
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