A new set of ephemerides for 15 of the largest asteroids has been produced for use in the Astronomical Almanac. The ephemerides cover the period from 1800 through 2100. The internal uncertainty in the mean longitude at epoch, 1997 December 18, ranges from 0.05" for 7 Iris through 0.22" for 65 Cybele, and the uncertainty in the mean motion varies from 0.02" per century for 4 Vesta to 0.14" per century for 511 Davida. This compares very favorably with the internal errors for the outer planets in recent Jet Propulsion Laboratory planetary ephemerides. However, because the asteroids have relatively little mass and are subject to perturbations by other asteroids, the actual uncertainties in their mean motions are likely to be a few tenths of an arcsecond per century. As part of the improvement to the ephemerides, new masses and densities were determined for 1 Ceres, 2 Pallas, and 4 Vesta, the three largest asteroids. These masses are as follows: Ceres=(4.39+/-0.04)x10^-10 M_solar, Pallas=(1.59+/-0.05)x10^-10 M_solar, and Vesta=(1.69+/-0.11)x10^-10 M_solar. The mass for Ceres is smaller than most previous determinations of its mass. This smaller mass is a direct consequence of the increase in the mass determined for Pallas. The densities found for these three asteroids are 2.00+/-0.03 g cm^-3 for Ceres, 4.2+/-0.3 gm cm^-3 for Pallas, and 4.3+/-0.3 g cm^-3 for Vesta. The density for Ceres is somewhat greater than that found for the taxonomically similar 253 Mathilde.