Variable emission from Sgr~A*, the luminous counterpart to the super-massive black hole at the center of our Galaxy, arises from the innermost portions of the accretion flow. Better characterization of the variability is important for constraining models of the low-luminosity accretion mode powering Sgr~A*, and could further our ability to use variable emission as a probe of the strong gravitational potential in the vicinity of the $4times10^{6}mathrm{M}_{odot}$ black hole. We use the textit{Herschel} Spectral and Photometric Imaging Receiver (SPIRE) to monitor Sgr~A* at wavelengths that are difficult or impossible to observe from the ground. We find highly significant variations at 0.25, 0.35, and 0.5 mm, with temporal structure that is highly correlated across these wavelengths. While the variations correspond to $<$1% changes in the total intensity in the textit{Herschel} beam containing Sgr~A*, comparison to independent, simultaneous observations at 0.85 mm strongly supports the reality of the variations. The lowest point in the light curves, $sim$0.5 Jy below the time-averaged flux density, places a lower bound on the emission of Sgr~A* at 0.25 mm, the first such constraint on the THz portion of the SED. The variability on few hour timescales in the SPIRE light curves is similar to that seen in historical 1.3 mm data, where the longest time series is available, but the distribution of variations in the sub-mm do not show a tail of large-amplitude variations seen at 1.3 mm. Simultaneous X-ray photometry from XMM-Newton shows no significant variation within our observing period, which may explain the lack of very large variations if X-ray and submillimeter flares are correlated.
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