  (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.
