We describe a series of experiments to characterize the sodium-layer guide star that was formed with the high-power laser developed for the Lawrence Livermore National Laboratory Atomic Vapor Laser Isotope Separation program. An emission spot size of 3.0 m was measured, with an implied laser irradiance spot diameter of 2.0 m. The rms spot motion at the higher laser powers, with active beam-pointing control, was less than 0.5 arcsec and had little effect on the observed spot size under these conditions. We measured the resonant backscatter from the sodium layer as a function of laser power to obtain a saturation curve. With a transmitted power of 1100 W and an atmospheric transmission of 0.6, the irradiance from the guide star at the ground was 10 (photons/cm(2))/ms, corresponding to a visual magnitude of 5.1. The implications for the performance of wave-front sensors with a laser guide star of this magnitude and resulting closed-loop adaptive-optics performance are discussed.