  Centaurus A (Cen A) is the nearest radio galaxy discovered as a
very-high-energy (VHE; 100 GeV-100 TeV) $\gamma$-ray source by the High Energy
Stereoscopic System (H.E.S.S.). It is a faint VHE $\gamma$-ray emitter, though
its VHE flux exceeds both the extrapolation from early Fermi-LAT observations
as well as expectations from a (misaligned) single-zone synchrotron-self
Compton (SSC) description. The latter satisfactorily reproduces the emission
from Cen A at lower energies up to a few GeV. New observations with H.E.S.S.,
comparable in exposure time to those previously reported, were performed and
eight years of Fermi-LAT data were accumulated to clarify the spectral
characteristics of the $\gamma$-ray emission from the core of Cen A. The
results allow us for the first time to achieve the goal of constructing a
representative, contemporaneous $\gamma$-ray core spectrum of Cen A over almost
five orders of magnitude in energy. Advanced analysis methods, including the
template fitting method, allow detection in the VHE range of the core with a
statistical significance of 12$\sigma$ on the basis of 213 hours of total
exposure time. The spectrum in the energy range of 250 GeV-6 TeV is compatible
with a power-law function with a photon index
$\Gamma=2.52\pm0.13_{\mathrm{stat}}\pm0.20_{\mathrm{sys}}$. An updated
Fermi-LAT analysis provides evidence for spectral hardening by
$\Delta\Gamma\simeq0.4\pm0.1$ at $\gamma$-ray energies above
$2.8^{+1.0}_{-0.6}$ GeV at a level of $4.0\sigma$. The fact that the spectrum
hardens at GeV energies and extends into the VHE regime disfavour a single-zone
SSC interpretation for the overall spectral energy distribution (SED) of the
core and is suggestive of a new $\gamma$-ray emitting component connecting the
high-energy emission above the break energy to the one observed at VHE
energies.
