We observed Sgr A* using the Very Large Array (VLA) and the Giant Metrewave Radio Telescope (GMRT) at multiple cm and mm wavelengths on 17 June 2003. The measured flux densities of Sgr A*, together with those obtained from the Submillimeter Array (SMA) and the Keck II 10 m telescope on the same date, are used to construct a simultaneous spectrum of Sgr A* from 90 cm to 3.8 micron . The simultaneous spectrum shows a spectral break at about 3.6 cm, a possible signature of synchrotron self-absorption of the strong radio outburst which occurred near epoch 17 July 2003. At 90 cm, the flux density of Sgr A* is 0.22+/-0.06 Jy, suggesting a sharp decrease in flux density at wavelengths longer than 47 cm. The spectrum at long cm wavelengths appears to be consistent with free-free absorption by a screen of ionized gas with a cutoff 100 cm. This cutoff wavelength appears to be three times longer than that of 30 cm suggested by Davies, Walsh & Booth based on observations in 1974 and 1975. Our analysis suggests that the flux densities of Sgr A* at wavelengths longer than 30 cm could be attenuated and modulated by stellar winds from massive stars close to Sgr A*.