Observations of nearby galaxy clusters at low surface brightness have identified galaxies with low luminosities, but sizes as large as L* galaxies, leading them to be dubbed ultra-diffuse galaxies (UDGs). The survival of UDGs in dense environments like the Coma cluster suggests that UDGs could reside in much more massive dark halos. We report the detection of a substantial population of globular clusters (GCs) around a Coma UDG, Dragonfly 17 (DF17). We find that DF17 has a high GC specific frequency of S_N=26+/-13. The GC system is extended, with an effective radius of 12+/-2, or 5.6+/-0.9 kpc at Coma distance, 70% larger than the galaxy itself. We also estimate the mean of the GC luminosity function to infer a distance of 97 (+17/-14) Mpc, providing redshift-independent confirmation that one of these UDGs is in the Coma cluster. The presence of a rich GC system in DF17 indicates that, despite its low stellar density, star formation was intense enough to form many massive star clusters. If DF17s ratio of total GC mass to total halo mass is similar to those in other galaxies, then DF17 has an inferred total mass of ~10^11 solar masses, only ~10% the mass of the Milky Way, but extremely dominated by dark matter, with M/L_V~1000. We suggest that UDGs like DF17 may be pure stellar halos, i.e., galaxies that formed their stellar halo components, but then suffered an early cessation in star formation that prevented the formation of any substantial central disk or bulge.
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