Non-commutative QED would lead to deviations from the Standard Model depending on a new energy scale Lambda_NC and a unique direction in space defined by two angles eta and xi. In this analysis, eta is defined as the angle between the unique direction and the rotation axis of the earth. The predictions of a tree level calculation for the process e^+e^- --> gamma gamma are evaluated for the specific orientation of the OPAL detector and compared to the measurements.Distributions of the polar and azimuthal photon angles are used to extract limits on the energy scale Lambda_NC depending on the model parameter eta. It is shown that the time dependence of the total cross-section could be used to determine the model parameter xi if there were a detectable signal. This is the first experimental study of non-commutative QED at an e^+e^- collider.