We have observed a bright flare of Sgr A* in the near infrared with the adaptive optics assisted integral field spectrometer SINFONI This work is based on observations collected at the European Southern Observatory, Paranal, Chile. Program ID: 075.B-0547(B). The observed spectrum is featureless and can be described by a power law. We show for the first time that the spectral index is correlated with the instantaneous flux and that both flux and spectral index experience significant changes within less than one hour. We argue that the near infrared flares from Sgr A* are due to synchrotron emission of transiently heated electrons, the emission being affected by orbital dynamics and synchrotron cooling, both acting on timescales of 20 minutes. The synchrotron cooling process may account for the observed variation in spectral index, which in turn requires a magnetic field 30 G, consistent with the equipartition field in a hot accretion flow with an accretion rate of 10-8 Mo/yr.