  It is widely believed that the bulk of the Galactic cosmic rays are
accelerated in supernova remnants (SNRs). However, no observational evidence of
the presence of particles of PeV energies in SNRs has yet been found. The young
historical SNR Cassiopeia A (Cas A) appears as one of the best candidates to
study acceleration processes. Between December 2014 and October 2016 we
observed Cas A with the MAGIC telescopes, accumulating 158 hours of
good-quality data. We derived the spectrum of the source from 100 GeV to 10
TeV. We also analysed $\sim$8 years of $Fermi$-LAT to obtain the spectral shape
between 60 MeV and 500 GeV. The spectra measured by the LAT and MAGIC
telescopes are compatible within the errors and show a clear turn off (4.6
$\sigma$) at the highest energies, which can be described with an exponential
cut-off at $E_c = 3.5\left(^{+1.6}_{-1.0}\right)_{\textit{stat}}
\left(^{+0.8}_{-0.9}\right)_{\textit{sys}}$ TeV. The gamma-ray emission from 60
MeV to 10 TeV can be attributed to a population of high-energy protons with
spectral index $\sim$2.2 and energy cut-off at $\sim$10 TeV. This result
indicates that Cas A is not contributing to the high energy ($\sim$PeV)
cosmic-ray sea in a significant manner at the present moment. A one-zone
leptonic model fails to reproduce by itself the multi-wavelength spectral
energy distribution. Besides, if a non-negligible fraction of the flux seen by
MAGIC is produced by leptons, the radiation should be emitted in a region with
a low magnetic field (B$\lessapprox$100$\mu$G) like in the reverse shock.)
