We present results from Hubble Space Telescope ultraviolet spectroscopy of the massive X-ray and black hole binary system, HD 226868 = Cyg X-1. The spectra were obtained at both orbital conjunction phases in 2002 and 2003, when the system was in the X-ray high/soft state. The UV stellar wind lines suffer large reductions in absorption strength when the black hole is in the foreground due to the X-ray ionization of the wind ions. We constructed model UV wind line profiles assuming that X-ray ionization occurs everywhere in the wind except the zone where the supergiant blocks the X-ray flux. The good match between the observed and model profiles indicates that the wind ionization extends to near the hemisphere of the supergiant facing the X-ray source. We also present contemporaneous spectroscopy of the Hα emission that forms in the high-density gas at the base of the supergiant's wind and the He II λ4686 emission that originates in the dense, focused wind gas between the stars. The Hα emission strength is generally lower in the high/soft state than in the low/hard state, but the He II λ4686 emission is relatively constant between X-ray states. The results suggest that mass transfer in Cyg X-1 is dominated by the focused wind flow that peaks along the axis joining the stars, and that the stellar wind contribution from the remainder of the hemisphere facing the X-ray source is shut down by X-ray photoionization effects (in both X-ray states). Based on observations with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. These observations are associated with programs GO-9646 and GO-9840. Based on data obtained at the David Dunlap Observatory, University of Toronto.