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Two years of microwave background observations with the Cosmic Background Imager (CBI) have been combined to give a sensitive, high-resolution angular power spectrum over the range 400< l< 3500. This power spectrum has been referenced to a more accurate overall calibration derived from the Wilkinson Microwave Anisotropy Probe (WMAP). The data cover 90 deg 2, including three pointings targeted for deep observations. The uncertainty on the l> 2000 power previously seen with the CBI is reduced. Under the assumption that any signal in excess of the primary anisotropy is due to a secondary Sunyaev-Zeldovich anisotropy in distant galaxy clusters, we use CBI, Arcminute Cosmology Bolometer Array Receiver, and Berkeley-Illinois-Maryland Association array data to place a constraint on the present-day rms mass fluctuation on 8 h-1 Mpc scales, σ 8. We present the results of a cosmological parameter …