We present measurements with the Very Long Baseline Array of the variability in the centroid position of Sgr A~* relative to a background quasar at 7 mm wavelength. We find an average centroid wander of 71 ±45 μas for time-scales between 50 and 100 minutes and 113 ± 50 μas for timescales between 100 and 200 minutes, with no secular trend. These are sufficient to begin constraining the viability of the accretion hot spot model for the radio variability of Sgr A~*. It is possible to rule out hot spots with orbital radii above 15GM_(Sgr A~*)/c~2 that contribute more than 30% of the total 7 mm flux. However, closer or less luminous hot spots remain unconstrained. Since the fractional variability of Sgr A~* during our observations was ～20% on timescales of hours, the hot spot model for Sgr A~*'s radio variability remains consistent with these limits. Improved monitoring of Sgr A~*'s centroid position has the potential to place significant constraints on the existence and morphology of inhomogeneities in a supermassive black hole accretion flow.
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