We report improved measurements of temperature anisotropies in the cosmic microwave background (CMB) radiation made with the Arcminute Cosmology Bolometer Array Receiver (ACBAR). In this paper, we use a new analysis technique and include 30% more data from the 2001 and 2002 observing seasons than the first release to derive a new set of band-power measurements with significantly smaller uncertainties. The planet-based calibration used previously has been replaced by comparing the flux of RCW 38 as measured by ACBAR and BOOMERANG to transfer the WMAP-based BOOMERANG calibration to ACBAR. The resulting power spectrum is consistent with the theoretical predictions for a spatially flat, dark energy-dominated ΛCDM cosmology, including the effects of gravitational lensing. Despite the exponential damping on small angular scales, the primary CMB fluctuations are detected with a signal-to-noise ratio of greater than 4, up to multipoles of l=2000. This increase in the precision of the fine-scale CMB power spectrum leads to only a modest decrease in the uncertainties on the parameters of the standard cosmological model. At high angular resolution, secondary anisotropies are predicted to be a significant contribution to the measured anisotropy. A joint analysis of the ACBAR results at 150 GHz and the CBI results at 30 GHz in the multipole range 2000<l<3000 shows that the power, reported by CBI in excess of the predicted primary anisotropy, has a frequency spectrum consistent with the thermal Sunyaev-Zel'dovich effect and inconsistent with primary CMB. The results reported here are derived from a subset of the total ACBAR data set; the final ACBAR power spectrum at 150 GHz will include 3.7 times more effective integration time and 6.5 times more sky coverage than is used here.