Galactic cosmic rays reach energies of at least a few Peta-electronvolts (1 PeV =$10^mathbf{15}$ electron volts). This implies our Galaxy contains PeV accelerators (PeVatrons), but all proposed models of Galactic cosmic-ray accelerators encounter non-trivial difficulties at exactly these energies. Tens of Galactic accelerators capable of accelerating particle to tens of TeV (1 TeV =$10^mathbf{12}$ electron volts) energies were inferred from recent gamma-ray observations. None of the currently known accelerators, however, not even the handful of shell-type supernova remnants commonly believed to supply most Galactic cosmic rays, have shown the characteristic tracers of PeV particles: power-law spectra of gamma rays extending without a cutoff or a spectral break to tens of TeV. Here we report deep gamma-ray observations with arcminute angular resolution of the Galactic Centre regions, which show the expected tracer of the presence of PeV particles within the central 10~parsec of the Galaxy. We argue that the supermassive black hole Sagittarius A* is linked to this PeVatron. Sagittarius A* went through active phases in the past, as demonstrated by X-ray outbursts and an outflow from the Galactic Centre. Although its current rate of particle acceleration is not sufficient to provide a substantial contribution to Galactic cosmic rays, Sagittarius A* could have plausibly been more active over the last $gtrsim 10^{6-7}$ years, and therefore should be considered as a viable alternative to supernova remnants as a source of PeV Galactic cosmic rays.
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