Cu-related green luminescence (GL) band in n-type ZnO layers grown by molecular beam epitaxy on sapphire and a bulk ZnO sample grown by hydrothermal method have been studied with above-bandgap excitation. Annealing of the samples in air at temperatures above 600 degrees C resulted in an increase in the concentration of Cu-Zn acceptors, followed by a dramatic enhancement of the Cu-related GL band peaking at 2.45 eV, and a characteristic fine structure. The GL band quenched at temperatures between 250 and 500 K due to escape of holes from the excited state of the Cu-Zn acceptor to the valence band. Energy position of this state in the bandgap of ZnO and its capture cross section for holes were estimated as 0.38 eV and 2x10(-13) cm(2), respectively.