We present the first detection from the Spitzer Space Telescope of 4.5 {mu}m variability from Sgr A*, the emitting source associated with the Milky Way's central black hole. The >23 hr continuous light curve was obtained with the Infrared Array Camera (IRAC) instrument in 2013 December. The result characterizes the variability of Sgr A* prior to the closest approach of the tidally deformed G2 object, a putative infalling gas cloud that orbits close to Sgr A*. The high stellar density at the location of Sgr A* produces a background of ~250 mJy at 4.5 {mu}m in each pixel with a large pixel-to-pixel gradient, but the light curve for the highly variable Sgr A* source was successfully measured by modeling and removing the variations due to pointing wobble. The observed flux densities range from the noise level of ~0.7 mJy rms in a 6.4 s measurement to >~10 mJy. Emission was seen above the noise level ~34% of the time. The light-curve characteristics, including the flux density distribution and structure function, are consistent with those previously derived at shorter infrared wavelengths. We see no evidence in the light curve for activity attributable to the G2 interaction at the observing epoch, ~100 days before the expected G2 periapsis passage. The IRAC light curve is more than a factor of two longer than any previous infrared observation, improving constraints on the timescale of the break in the power spectral distribution of Sgr A* flux densities. The data favor the longer of the two previously published values for the timescale.