I present a catalog of 67 barred galaxies which contain distinct, elliptical stellar structures inside their bars. Fifty of these are double-barred galaxies: a small-scale, inner or secondary bar is embedded within a large-scale, outer or primary bar. I provide homogenized measurements of the sizes, ellipticities, and orientations of both inner and outer bars, along with global parameters for the galaxies. The other 17 are classified as inner-disk galaxies, where a large-scale bar harbors an inner elliptical structure which is aligned with the galaxy's outer disk. Four of the double-barred galaxies also possess inner disks, located in between the inner and outer bars. While the inner-disk classification is ad-hoc - and undoubtedly includes some inner bars with chance alignments (five such probable cases are identified) - there is good evidence that inner disks form a statistically distinct population, and that at least some are indeed disks rather than bars. In addition, I list 36 galaxies which may be double-barred, but for which current observations are ambiguous or incomplete, and another 23 galaxies which have been previously suggested as potentially being double-barred, but which are probably not. False double-bar identifications are usually due to features such as nuclear rings and spirals being misclassified as bars; I provide some illustrated examples of how this can happen. A detailed statistical analysis of the general population of double-bar and inner-disk galaxies, as represented by this catalog, will be presented in a companion paper. Tables ef{tab:measured} and ef{tab:deproj} are only available in electronic form at http://www.edpsciences.org