The real-space structure of hard-sphere glasses quenched from colloidal liquids in thermodynamic equilibrium has been determined. Particle coordinates were obtained by combining the optical sectioning capability of confocal fluorescence microscopy with the structure of specially prepared fluorescent silica colloids. Both the average structure and the local structure of glasses, with volume fractions ranging from 0.60 to 0.64, were in good agreement with glasses and random close packings generated by computer simulations. No evidence of a divergent correlation length was found. The method used to obtain the three-dimensional particle coordinates is directly applicable to other colloidal structures, such as crystals, gels, and flocs.