We analyze the physical conditions in the low-ionization component of starburst outflows (in contrast to the high-ionization wind fluid observed in X-rays), based on new Keck/LRIS spectroscopy of partially resolved absorption troughs in near-ultraviolet and optical spectra of Ultraluminous Infrared Galaxies. The large velocity width and blueshift present in seven atomic transitions indicate a macroscopic velocity gradient within the outflowing gas. The separation of the Mg II 2796, 2803 (and Fe II 2587, 2600) doublet lines in these data constrains the gas kinematics better than previous studies of the heavily blended Na I 5892, 5898 doublet. The identical shape of the Mg II 2796 absorption troughs to that of the normally weaker transition at 2803 Å (after accounting for emission filling) requires both transitions be optically thick at all outflow velocities. The fraction of the galactic continuum covered by the outflow at each velocity therefore dictates the shape of these absorption troughs. We suggest that the velocity offset of the deepest part of the troughs, where the covering factor of low-ionization gas is near unity, reflects the speed of a shell of swept-up, interstellar gas at the time of blowout. In a spherical outflow, we show that the fragments of this shell, or any clouds that expand adiabatically in rough pressure equilibrium with the hot wind, expand slowly relative to the geometrical dilution, and the covering fraction of low-ionization gas decreases with increasing radius. Our measurement of a covering factor that decreases with increasing velocity can therefore be interpreted as evidence that the low-ionization outflow is accelerating, i.e., absorption at higher velocity comes from gas at larger radii. We also present measurements of C_f (v) in four species, place an upper limit of n_e 3000 cm^–3 on the density of the outflowing gas, and discuss lower limits on the mass outflow rate.