See paper for full list of authors - Manuscript submitted to A&A ; Aims. The gamma-ray emission from the shell-type supernova remnant (SNR) RX J0852.0-4622 is studied in order to better characterize its spectral properties and its distribution over the SNR.Methods. The analysis of an extended H.E.S.S. data set at very-high energies (E > 100 GeV) permits detailed studies of the morphology and the spectrum of the whole RX J0852.0-4622 region, as well as spatially-resolved spectroscopy. 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 cut-off at an energy of Ecut = (6.7 +/- 1.2_stat +/- 1.2_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 cut-off. 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.