3C 351 (z = 0.371), an X-ray-''quiet'' quasar, is one of the few quasars showing signs of a ''warm absorber'' in its X-ray spectrum; i.e., partially ionized absorbing material in the line of sight whose opacity depends on its ionization structure. The main feature in the X-ray spectrum is a K-edge due to O VII or O VIII. 3C 351 also shows unusually strong, blueshifted, associated, absorption lines in the ultraviolet (Bahcall et al. 1993) including O VI (lambdalambda1031, 1037). This high ionization state strongly suggests an identification with the X-ray absorber and a site within the active nucleus. In this paper we demonstrate that the X-ray and UV absorption is due to the same material. This is the first confirmed UV/X-ray absorber. Physical conditions of the absorber are determined through the combination of constraints derived from both the X-ray and UV analysis. This highly ionized, outflowing, low-density, high-column density absorber situated outside the broad emission line region (BELR) is a previously unknown component of nuclear material. We rule out the identification of the absorber with a BELR cloud as the physical conditions in the two regions are inconsistent with one another. The effect of the X-ray quietness and IR upturn in the 3C 351 continuum on the BELR is also investigated. The strengths of the high-ionization lines of C IV lambda1549 and O VI lambda1034 with respect to Lyalpha are systematically lower (up to a factor of 10) in the material ionized by the 3C 351 continuum as compared to those produced by the ''standard'' quasar continuum, the strongest effect being on the strength of O VI lambda1034. We find that for a 3C 351-like continuum, C III] lambda1909 ceases to be a density indicator.