We present echelle observations of the intrinsic UV absorption lines in the Seyfert galaxy NGC 4151, which were obtained with the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope on 1999 July 19. The UV continuum flux at 1450 Angstroms decreased by factor of about four over the previous two years and there was a corresponding dramatic increase in the column densities of the low-ionization absorption lines (e.g., Si II, Fe II, and Al II), presumably as a result of a decrease in the ionizing continuum. In addition to the absorption lines seen in previous low states, we identify a large number of Fe II absorption lines that arise from metastable levels as high as 4.1 eV above the ground state, indicating high densities (> 10^6 cm^-3). We find that the transient absorption feature in the blue wing of the broad C IV emission, seen in a Goddard High Resolution Spectrograph spectrum and thought to be a high-velocity C~IV component, is actually a Si II fine-structure absorption line at a radial velocity of -560 km/s (relative to systemic). We also demonstrate that the ``satellite emission lines of C~IV found in International Ultraviolet Explorer spectra are actually regions of unabsorbed continuum plus broad emission that become prominent when the UV continuum of NGC 4151 is in a low state.