We present near-infrared spectroscopy for a complete sample of 33 ultraluminous infrared galaxies at a resolution of R\approx 1000. Most of the wavelength range from 1.80-2.20 microns in the rest frame is covered, including the Pa-alpha and Br-gamma hydrogen recombination lines, and the molecular hydrogen vibration-rotation 1-0 S(1) and S(3) lines. Other species, such as He I, [Fe II], and [Si VI] appear in the spectra as well, in addition to a number of weaker molecular hydrogen lines. Nuclear extractions for each of the individual galaxies are presented here, along with spectra of secondary nuclei, where available. The Pa-alpha emission is seen to be highly concentrated on the nuclei, typically with very little emision extending beyond a radius of 1 kpc. Signatures of active nuclei are rare in the present sample, occurring in only two of the 33 galaxies. It is found that visual extinctions to the nuclei via the Pa-alpha/Br-gamma line ratio in excess of 10 magnitudes are relatively common among ULIRGs, and that visual extinctions greater than 25 mag are necessary to conceal a QSO emitting half the total bolometric luminosity. The vibration-rotation lines of molecular hydrogen appear to be predominantly thermal in origin, with effective temperatures generally around 2200 K. The relative nuclear velocities between double nucleus ULIRGs are investigated, through which it is inferred that the maximum deprojected velocity difference is about 200 km/s. This figure is lower than the velocities predicted by physical models of strong interactions/mergers of large, gas-rich galaxies.

52 pages (19 with just figures), 9 figures, accepted for publication in the Astronomical Journal; Table 3 not formatted properly on astro-ph: get source and print Murphy.tab3.ps