CO (1 --> 0) emission profiles are presented for 60 infrared luminous, [L(ir)(8-1000-mu-m) = 5 x 10(10)-3 x 10(12) L., H0 = 75 km s-1 Mpc-1], bright (F60-mu-m > 5.24 Jy), IRAS galaxies at redshifts 1500-25,000 km s-1. Most of these galaxies are sufficiently distant that a single 55" beam measures the total CO luminosity and hence the total molecular gas content. The data for these galaxies are analyzed together with that for a sample of 29 closer IRAS bright galaxies with L(ir) down to 2 x 10(10) L. mapped by Tinney et al. (1990). All of the IRAS bright galaxies are extremely rich in molecular gas with M(H-2) = 1 x 10(9)-6 x 10(10) M., corresponding to 0.5-30 times the mass of H-2 gas in the Milky Way. The dust masses are evaluated from the IRAS data and the mean ratio of H-2 gas to "warm dust" is 540 +/- 290. Including the mass of heavy elements and the contribution from H I implies a mean gas-to-dust ratio in the range 900-1100. The discrepancy between this value and the standard Galactic gas-to-dust ratio of approximately 150 could be due to the presence of large quantities of cold dust to which IRAS is insensitive, and/or errors in the assumed dust mass absorption coefficient. The infrared luminosity-to-H-2 mass ratios, L(ir)/M(H-2), range from 2-220 L. M.-1, up to a factor of 55 times the mean ratio for molecular clouds in the Galaxy. The ratio L(ir)/M(H-2) increases with increasing L(ir), but shows no systematic dependence on M(H-2). The higher values of L(ir)/M(H-2) obtained in the most luminous galaxies suggests either an increased star formation efficiency and/or an additional source of luminosity. The kinetic energy of colliding galaxies is ruled out as a significant source of the luminosity in view of the independence of the L(ir)/M(H-2) ratios on the galactic velocity dispersions and the merging time scales which are too long. If the excess luminosity is supplied by star formation, the implied lifetime of the interstellar medium is less-than-or-equal-to 3 x 10(8) yr. However, a significant number of the galaxies with the highest L(ir)/M(H-2) ratios exhibit nonthermal optical emission lines characteristic of an active galactic nucleus (AGN), and an AGN appears to be an important energy source for these galaxies.