We report near-simultaneous interferometric measurements of the spectrum of Sgr A* over the 5 to 354 GHz range, and single-dish observations which have yielded the first detection of Sgr A* at 850 GHz. We confirm that Sgr A*'s spectrum rises more steeply at short mm wavelengths than at cm wavelengths, leading to a near-millimeter/submillimeter excess which dominates its luminosity. Below 900 GHz, Sgr A*'s observed luminosity is 70 +/- 30 L_o. A new upper limit to Sgr A*'s 24.3 micron flux, together with a compilation of other extant IR data, imply a FIR spectral turnover, which can result either from an intrinsic synchrotron cutoff, or excess extinction near Sgr A^*. If the former applies, Sgr A*'s total synchrotron luminosity is < 10^3 L_o, while in the latter case it is <3 * 10^4 L_o if spherical symmetry also applies.