THE morphology of our Galaxy is characterized by a disk of stars moving on circular orbits, surrounding a central spheroidal body of stars on high-velocity, randomly oriented orbits. The spheroid is further differentiated into an inner bulge and an outer halo; the bulge stars are rich in elements heavier than helium (’metals’), whereas the halo stars are metal-poor, suggesting that the latter formed very early in the history of the Galaxy. (They have experienced little chemical enrichment, by previous generations of stars.) It is not known, however, whether the bulge is the inner extension of the halo, having formed as part of the same process(1), or whether it formed much later, perhaps by a dynamical distortion of the inner regions of the disk(2,3). Here we report observations obtained with the Hubble Space Telescope of two metal-rich globular clusters that form part of the bulge population. Within the uncertainties, these bulge globular clusters appear to be coeval with halo clusters, which suggests that the formation of the bulge was part of the dynamical process that formed the halo, and that the bulge gas underwent rapid chemical enrichment, in less than a few billion years.