  We present new Hubble Space Telescope (HST)/Cosmic Origins Spectrograph
observations of the Narrow-Line Seyfert 1 galaxy NGC 4051. These data were
obtained as part of a coordinated observing program including X-ray
observations with the Chandra/High Energy Transmission Grating (HETG)
Spectrometer and Suzaku. We detected nine kinematic components of UV
absorption, which were previously identified using the HST/Space Telescope
Imaging Spectrograph. None of the absorption components showed evidence for
changes in column density or profile within the \sim 10 yr between the STIS and
COS observations, which we interpret as evidence of 1) saturation, for the
stronger components, or 2) very low densities, i.e., n_H < 1 cm^-3, for the
weaker components. After applying a +200 km s^-1 offset to the HETG spectrum,
we found that the radial velocities of the UV absorbers lay within the O VII
profile. Based on photoionization models, we suggest that, while UV components
2, 5 and 7 produce significant O VII absorption, the bulk of the X-ray
absorption detected in the HETG analysis occurs in more highly ionized gas.
Moreover, the mass loss rate is dominated by high ionization gas which lacks a
significant UV footprint.
