Aims. We study gamma-ray emission from the shell-type supernova remnant (SNR) RX J0852.0$-$4622 to better characterize its spectral properties and its distribution over the SNR. Methods. The analysis of an extended High Energy Spectroscopic System (H.E.S.S.) data set at very high energies (E > 100 GeV) permits detailed studies, as well as spatially resolved spectroscopy, of the morphology and spectrum of the whole RX J0852.0$-$4622 region. The H.E.S.S. data are combined with archival data from other wavebands and interpreted in the framework of leptonic and hadronic models. The joint Fermi-LAT-H.E.S.S. spectrum allows the direct determination of the spectral characteristics of the parent particle population in leptonic and hadronic scenarios using only GeV-TeV data. Results. An updated analysis of the H.E.S.S. data shows that the spectrum of the entire SNR connects smoothly to the high-energy spectrum measured by Fermi-LAT. The increased data set makes it possible to demonstrate that the H.E.S.S. spectrum deviates significantly from a power law and is well described by both a curved power law and a power law with an exponential cutoff at an energy of $E_mathrm{cut} = (6.7 pm 1.2_mathrm{stat} pm 1.2_mathrm{syst})$ TeV. The joint Fermi-LAT-H.E.S.S. spectrum allows the unambiguous identification of the spectral shape as a power law with an exponential cutoff. No significant evidence is found for a variation of the spectral parameters across the SNR, suggesting similar conditions of particle acceleration across the remnant. A simple modeling using one particle population to model the SNR emission demonstrates that both leptonic and hadronic emission scenarios remain plausible. It is also shown that at least a part of the shell emission is likely due to the presence of a pulsar wind nebula around PSR J0855$-$4644.
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