We present observations of the UV absorption lines in the Seyfert 1 galaxy NGC 3516, obtained at a resolution of =D % 40; 000 with the Space Telescope Imaging Spectrograph (STIS) on 2000 October 1. The UV continuum was $4 times lower than that observed during 1995 with the Goddard High Resolution Spectrograph (GHRS), and the X-ray flux from a contemporaneous Chandra X-Ray Observatory observation was a factor of $8 below that observed with ASCA. The STIS spectra show kinematic components of absorption in Ly , C iv, and N v at radial velocities of À376, À183, and À36 km s À1 (components 1, 2, and 3+4, respectively), which were detected in the earlier GHRS spectra; the last of these is a blend of two GHRS components that have increased greatly in column density. Four additional absorption components have appeared in the STIS spectra at radial velocities of À692, À837, À994, and À1372 km s À1 (components 5-8); these may also have been present in earlier low-flux states observed by the International Ultraviolet Explorer. Based on photoionization models, we suggest that the components are arranged in increasing radial distance in the order 3+4, 2, 1, followed by components 5-8. We have achieved an acceptable fit to the X-ray data using the combined X-ray opacity of the UV components 1, 2, and 3+4. By increasing the UV and X-ray fluxes of these models to match the previous high states, we are able to match the GHRS C iv column densities, the absence of detectable C iv absorption in components 5-8, and the 1994 ASCA spectrum. We conclude that variability of the UV and X-ray absorption in NGC 3516 is primarily due to changes in the ionizing flux.