We report on the first experimental characterization of a gamma-ray spectrometer designed to spectrally resolve high-flux photon beams with energies in the GeV range. The spectrometer has been experimentally characterized using a bremsstrahlung source obtained at the Apollon laser facility during the interaction of laser-wakefield accelerated electron beams (maximum energy of 1.7 GeV and overall charge of 207±62 pC) with a 1 mm thick tantalum target. Experimental data confirms the possibility of performing single-shot measurements, without the need for accumulation, with a high signal-to-noise ratio. Scaling the results to photons in the multi-GeV range suggests the possibility of achieving percent-level energy resolution as required, for instance, by the next generation of experiments in strong-field quantum electrodynamics.