We present here a novel scheme for the high-resolution spectrometry of high-flux gamma-ray beams with energies per photon in the multi-GeV range. The spectrometer relies on the conversion of the gamma-ray photons into electron-positron pairs in a solid foil with high atomic number. The measured electron and positron spectra are then used to reconstruct the spectrum of the gamma-ray beam. The performance of the spectrometer has been numerically tested against the predicted photon spectra expected from non-linear Compton scattering in the proposed LUXE experiment, showing high fidelity in identifying distinctive features such as Compton edges and non-linearities.