We have used the Canada-France-Hawaii Telescope on Mauna Kea to obtain two deep CCD exposures in the central region of the Coma cluster, and we have used the new images to detect globular clusters in the central giant galaxy NGC 4874. One exposure with a total integration time of 2 hours lies in the halo of the giant galaxy. A second (comparison) exposure of the same integration time was taken within the cluster core but well away from any other bright galaxy. Our analysis, which includes both photometry and image classification with the automated routines called FOCAS, extends to a limiting B magnitude of 26.25. Image classification is a necessary part of this study because at the faintest limiting magnitudes globular clusters are easily confused with the rich population of faint background galaxies. We have been particularly successful in sorting stellar objects from galaxies because the image quality on our images is quite good: FWHM ~ 0.6 arcsec. The data were taken with the fast autoguiding instrument called ISIS (Thompson and Ryerson 1984)