(Abridged) Context. NGC 253 is one of only two starburst galaxies found to emit $gamma$-rays from hundreds of MeV to multi-TeV energies. Accurate measurements of the very-high-energy (VHE) (E $>$ 100 GeV) and high-energy (HE) (E $>$ 60 MeV) spectra are crucial to study the underlying particle accelerators and cosmic-ray interaction and transport. Aims. The measurement of the VHE $gamma$-ray emission of NGC 253 published in 2012 by H.E.S.S. was limited by large systematic uncertainties. Here, a measurement of the $gamma$-ray spectrum of NGC 253 is investigated in both HE and VHE $gamma$-rays. Methods. The data of H.E.S.S. observations are reanalysed using an updated calibration and analysis chain. The $Fermi$-LAT analysis employs more than 8 years of data processed using pass 8. The cosmic-ray particle population is evaluated from the combined HE--VHE $gamma$-ray spectrum using NAIMA. Results. The VHE $gamma$-ray energy spectrum is best fit by a power-law with a flux normalisation of $(1.34,pm,0.14^{mathrm{stat}},pm,0.27^{mathrm{sys}}) times 10^{-13} mathrm{cm^{-2} s^{-1} TeV^{-1}}$ at 1 TeV -- about 40 % above, but compatible with the value obtained in Abramowski et al. (2012). The spectral index $Gamma = 2.39 pm 0.14^{mathrm{stat}} pm 0.25^{mathrm{sys}}$ is slightly softer than but consistent with the previous measurement. At energies above $sim$3 GeV the HE spectrum is consistent with a power-law ranging into the VHE part of the spectrum measured by H.E.S.S. Conclusions. Two scenarios for the starburst nucleus are tested, in which the gas in the starburst nucleus acts as a target for hadronic cosmic rays. In these two models, the level to which NGC,253 acts as a calorimeter is estimated to a range of $f_{rm cal} = 0.1$ to $1$ while accounting for the measurement uncertainties.
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