We present H-0 results from Cosmic Background Imager ( CBI) observations of the Sunyaev-Zel'dovich effect (SZE) in seven galaxy clusters, A85, A399, A401, A478, A754, A1651, and A2597. These observations are part of a program to study a complete, volume-limited sample of low-redshift (z < 0.1), X-ray-selected clusters. Our focus on nearby objects allows us to study a well-defined, orientation-unbiased sample, minimizing systematic errors due to cluster asphericity. We use density models derived from ROSAT imaging data and temperature measurements from ASCA and BeppoSAX spectral observations. We quantify in detail sources of error in our derivation of H-0, including calibration of the CBI data, density, and temperature models from the X-ray data, cosmic microwave background primary anisotropy fluctuations, and residuals from radio point source subtraction. From these seven clusters we obtain a result of H-0 = 67(-18-6)(+30+15) km s(-1) Mpc(-1) for an unweighted sample average. The respective quoted errors are random and systematic uncertainties at 68% confidence. The dominant source of error is confusion from intrinsic anisotropy fluctuations.