  We present the first fully simultaneous fits to the NIR and X-ray spectral
slope (and its evolution) during a very bright flare from Sgr A*, the
supermassive black hole at the Milky Way's center. Our study arises from
ambitious multi-wavelength monitoring campaigns with XMM-Newton, NuSTAR and
SINFONI. The average multi-wavelength spectrum is well reproduced by a broken
power-law with $\Gamma_{NIR}=1.7\pm0.1$ and $\Gamma_X=2.27\pm0.12$. The
difference in spectral slopes ($\Delta\Gamma=0.57\pm0.09$) strongly supports
synchrotron emission with a cooling break. The flare starts first in the NIR
with a flat and bright NIR spectrum, while X-ray radiation is detected only
after about $10^3$ s, when a very steep X-ray spectrum
($\Delta\Gamma=1.8\pm0.4$) is observed. These measurements are consistent with
synchrotron emission with a cooling break and they suggest that the high energy
cut-off in the electron distribution ($\gamma_{max}$) induces an initial
cut-off in the optical-UV band that evolves slowly into the X-ray band. The
temporal and spectral evolution observed in all bright X-ray flares are also in
line with a slow evolution of $\gamma_{max}$. We also observe hints for a
variation of the cooling break that might be induced by an evolution of the
magnetic field (from $B\sim30\pm8$ G to $B\sim4.8\pm1.7$ G at the X-ray peak).
Such drop of the magnetic field at the flare peak would be expected if the
acceleration mechanism is tapping energy from the magnetic field, such as in
magnetic reconnection. We conclude that synchrotron emission with a cooling
break is a viable process for Sgr A*'s flaring emission.
