We confirm that the object DDO 216-A1 is a substantial globular cluster at the center of Local Group galaxy DDO 216 (the Pegasus dwarf irregular), using Hubble Space Telescope ACS imaging. By fitting isochrones, we find the cluster metallicity [M/H]=-1.6+/-0.2, for reddening E(B-V)=0.16+/-0.02; the best-fit age is 12.3+/-0.8Gyr. There are ~30 RR Lyrae variables in the cluster; the magnitude of the fundamental mode pulsators gives a distance modulus of 24.77+/-0.08 --identical to the host galaxy. The ratio of overtone to fundamental mode variables and their mean periods make DDO 216-A1 an Oosterhoff Type I cluster. We find a central surface brightness of 20.85+/-0.17 F814W mag/arcsec^2^, a half-light radius of 3.1" (13.4pc), and an absolute magnitude M814=-7.90+/-0.16 (M/M_{sun}_~10^5^). King models fit to the cluster give the core radius and concentration index, r_c_=2.1"+/-0.9" and c=1.24+/-0.39. The cluster is an "extended" cluster somewhat typical of some dwarf galaxies and the outer halo of the Milky Way. The cluster is projected <~30pc south of the center of DDO 216, unusually central compared to most dwarf galaxy globular clusters. Analytical models of dynamical friction and tidal destruction suggest that it probably formed at a larger distance, up to ~1kpc, and migrated inward. DDO 216 has an unexceptional specific cluster frequency, S_N_=10. DDO 216 is the lowest-luminosity Local Group galaxy to host a 10^5^M_{sun}_ globular cluster and the only transition-type (dSph/dIrr) galaxy in the Local Group with a globular cluster.