The amplitude of the magnetic field near the Galactic Centre has been uncertain by two orders of magnitude for several decades. On a scale of approximately 100 parsecs (pc), fields of approximately 1,000 microgauss (microG; refs 1-3) have been reported, implying a magnetic energy density more than 10,000 times stronger than typical for the Galaxy. Alternatively, the assumption of pressure equilibrium between the various phases of the Galactic Centre interstellar medium (including turbulent molecular gas, the contested 'very hot' plasma, and the magnetic field) suggests fields of approximately 100 microG over approximately 400 pc size scales. Finally, assuming equipartition, fields of only approximately 6 microG have been inferred from radio observations for 400 pc scales. Here we report a compilation of previous data that reveals a downward break in the region's non-thermal radio spectrum (attributable to a transition from bremsstrahlung to synchrotron cooling of the in situ cosmic-ray electron population). We show that the spectral break requires that the Galactic Centre field be at least approximately 50 microG on 400 pc scales, lest the synchrotron-emitting electrons produce too much gamma-ray emission, given other existing constraints. Other considerations support a field of 100 microG, implying that over 10% of the Galaxy's magnetic energy is contained in only less than or approximately 0.05% of its volume.