Results are reported from an all-sky survey carried out at high X-ray energies (13-180 keV) from August 1977 until January 1979 using data obtained with the UCSD/MIT Hard X-Ray and Low-Energy Gamma-Ray Instrument on the HEAO 1 satellite. Visual displays are presented which indicate qualitatively the location, intensities, and time variability of the detected high-energy X-ray sources. A model-dependent procedure for the quantitative analysis of the sky survey data is described. The results of this procedure are presented in tabular form and include fitted count rates in four broad energy bands for about 70 sources. All sources which were detected at a level of statistical significance of not less than about 6 sigma were clearly evident in the visual displays of sky survey data. The survey is therefore complete, except in regions of source confusion, down to an intensity level of about 1/75 of the Crab Nebula in the 13-80 keV band. Forty-four sources were detected in the 40-80 keV energy band, and 14 were detected in the 80-180 keV band. Although most of the detected sources are galactic, seven are extragalactic.