We have observed the metal-rich globular clusters NGC 6624 and NGC 6637 (M69) using the planetary camera of the WFPC2 on the Hubble Space Telescope (HST). Observations of the Ca II triplet lines in giant stars in these clusters show that NGC 6624 and NGC 6637 have metallicities on the Zinn and West scale of [Fe/H]=-0.63+/-0.09 and -0.65+/-0.09, only slightly more metal rich than 47 Tuc [Fe/H]=-0.71+/-0.07. For clusters of identical (or nearly so) metallicity, one can make a direct comparison of the color-magnitude diagrams to derive the relative ages of the clusters. From the color-magnitude diagrams derived from the HST photometry, we find that NGC 6624 and NGC 6637 differ in age by less than 0.5 Gyr. Their color-magnitude diagrams are also compared with those of 47 Tuc and NGC 6352, and while these latter diagrams are of somewhat lower quality, they are consistent with all of these clusters having the same ages. Adopting an apparent distance modulus of 13.40 and reddening E(B-V)=0.04 for 47 Tuc, the new Yale isochrones yield an age for the clusters of 14 Gyr. The positions of NGC 6624 and NGC 6637 in the Galaxy suggest that they belong to the bulge population of globular clusters. The only other bulge clusters that have been dated so far are the more metal rich clusters NGC 6528 and NGC 6553, which also appear to be very old. Consequently, the age-metallicity relation of the bulge may be very steep. The close similarity of the ages and metallicities of NGC 6624 and NGC 6637 to the thick-disk globular clusters 47 Tuc and NGC 6352 indicates that the age-metallicity relations of these populations intersect. We briefly discuss the possibility that these populations had a common origin. Based on observations made with the Anglo-Australian Telescope, Canada-France-Hawaii Telescope, and the NASA/ESA Hubble Space Telescope. The observations from the Hubble Space Telescope were obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.