We present an optical eclipse observation of the hot Jupiter WASP-12b using the Space Telescope Imaging Spectrograph on board the Hubble Space Telescope. These spectra allow us to place an upper limit of A(g) < 0.064 (97.5% confidence level) on the planet's white light geometric albedo across 290-570 nm. Using six wavelength bins across the same wavelength range also produces stringent limits on the geometric albedo for all bins. However, our uncertainties in eclipse depth are similar to 40% greater than the Poisson limit and may be limited by the intrinsic variability of the Sun-like host star-the solar luminosity is known to vary at the 10(-4) level on a timescale of minutes. We use our eclipse depth limits to test two previously suggested atmospheric models for this planet: Mie scattering from an aluminum-oxide haze or cloud-free Rayleigh scattering. Our stringent nondetection rules out both models and is consistent with thermal emission plus weak Rayleigh scattering from atomic hydrogen and helium. Our results are in stark contrast with those for the much cooler HD 189733b, the only other hot Jupiter with spectrally resolved reflected light observations; those data showed an increase in albedo with decreasing wavelength. The fact that the first two exoplanets with optical albedo spectra exhibit significant differences demonstrates the importance of spectrally resolved reflected light observations and highlights the great diversity among hot Jupiters.