We present observations of the UV absorption lines in the Seyfert 1 galaxy NGC 3516, obtained at a resolution of λ/Δλ ≈ 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 Lyα, C IV, and N V at radial velocities of -376, -183, and -36 km s-1 (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 -692, -837, -994, and -1372 km s-1 (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.