Few published ultraviolet (UV) spectra exist for stripped-envelope supernovae and none to date for broad-lined Type Ic supernovae (SNe Ic-bl). These objects have extremely high ejecta velocities and are the only supernova type directly linked to gamma-ray bursts (GRBs). Here we present two epochs of HST/STIS spectra of the SN Ic-bl 2014ad, the first UV spectra for this class. We supplement this with 26 new epochs of ground-based optical spectra, augmenting a rich spectral time series. The UV spectra do not show strong features and are consistent with broadened versions of other SN Ic spectra observed in the UV. We measure Fe II 5169 Å velocities and show that SN 2014ad has even higher ejecta velocities than most SNe Ic both with and without observed GRBs. We construct models of the SN 2014ad UV+optical spectra using TARDIS, a 1D Monte Carlo radiative-transfer spectral synthesis code. The models fit the data well at multiple epochs in the optical but underestimate the flux in the UV, likely due to simplifying assumptions. We find that high densities at high velocities are needed to reproduce the spectra, with ~3 M ⊙ of material at v > 22,000 km s-1, assuming spherical symmetry. Our nebular line fits suggest a steep density profile at low velocities. Together, these results imply a higher total ejecta mass than estimated from previous light-curve analysis and expected from theory. This may be reconciled by a flattening of the density profile at low velocity and extra emission near the center of the ejecta.