A Fast Radio Burst in a Compact Galaxy Group at $z$~1

FRB 20220610A is a high-redshift Fast Radio Burst (FRB) that has not been observed to repeat. Here, we present rest-frame UV and optical $\textit{Hubble Space Telescope}$ observations of the field of FRB 20220610A. The imaging reveals seven extended sources, one of which we identify as the most likely host galaxy with a spectroscopic redshift of $z$=1.017. We spectroscopically confirm at least three additional sources to be at the same redshift, and identify the system as a compact galaxy group with possible signs of interaction among group members. We determine the host of FRB 20220610A to be a star-forming galaxy with stellar mass of $\approx10^{9.7}\,M_{\odot}$, mass-weighted age of $\approx2.6$~Gyr, and star formation rate (integrated over the last 100 Myr) of $\approx1.7$~M$_{\odot}$~yr$^{-1}$. These host properties are commensurate with the star-forming field galaxy population at z~1 and trace their properties analogously to the population of low-$z$ FRB hosts. Based on estimates of the total stellar mass of the galaxy group, we calculate a fiducial contribution to the observed Dispersion Measure (DM) from the intragroup medium of $\approx 110-220$ $\rm pc \, cm^{-3}$ (rest-frame). This leaves a significant excess of $500^{+272}_{-109}$ $\rm pc \, cm^{-3}$ (in the observer frame), with additional sources of DM possibly originating from the circumburst environment, host galaxy interstellar medium, and/or foreground structures along the line of sight. Given the low occurrence rates of galaxies in compact groups, the discovery of an FRB in such a group demonstrates a rare and novel environment in which FRBs can occur.