  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 pc fields of approximately 1000 microG 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 down-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 50
microG on 400 pc scales, lest the synchrotron-emitting electrons produce too
much gamma-ray emission given existing constraints. Other considerations
support a field of 100 microG, implying that > 10% of the Galaxy's magnetic
energy is contained in only < 0.05% of its volume.
