Using the new Cosmic Origins Spectrograph (COS) on the {\it Hubble Space Telescope (HST)}, we obtained moderate-resolution, high signal/noise ultraviolet spectra of HD 209458 and its exoplanet HD 209458b during transit, both orbital quadratures, and secondary eclipse. We compare transit spectra with spectra obtained at non-transit phases to identify spectral features due to the exoplanet's expanding atmosphere. We find that the mean flux decreased by $7.8± 1.3$% for the C II 1334.5323Å\ and 1335.6854Å\ lines and by $8.2± 1.4$% for the Si III 1206.500Å\ line during transit compared to non-transit times in the velocity interval --50 to +50 km s$^{-1}$. Comparison of the C II and Si III line depths and transit/non-transit line ratios shows deeper absorption features near --10 and +15 km s$^{-1}$ and less certain features near --40 and +30--70 km s$^{-1}$, but future observations are needed to verify this first detection of velocity structure in the expanding atmosphere of an exoplanet. Our results for the C II lines and the non-detection of Si IV 1394.76Å\ absorption are in agreement with \citet{Vidal-Madjar2004}, but we find absorption during transit in the Si III line contrary to the earlier result. The $8± 1$% obscuration of the star during transit is far larger than the 1.5% obscuration by the exoplanet's disk. Absorption during transit at velocities between --50 and +50 km s$^{-1}$ in the C II and Si III lines requires high-velocity ion absorbers, but models that assume that the absorbers are high-temperature thermal ions are inconsistent with the COS spectra. Assuming hydrodynamic model values for the gas temperature and outflow velocity at the limb of the outflow as seen in the C II lines, we find mass-loss rates in the range (8--40)$\times 10^{10}$ g s$^{-1}$. ; Comment: 25 pages, 4 figures, Astrophysical Journal in press