We apply near-threshold laser photodetachment to characterize the rotational quantum level distribution of OH$^-$ ions stored in the cryogenic ion-beam storage ring, DESIREE, at Stockholm University. We find that the stored ions relax to a rotational temperature of 13.4$\pm$0.2 K with 94.9$\pm$0.3 % of the ions in the rotational ground state. This is consistent with the storage ring temperature of 13.5$\pm$0.5 K as measured with eight silicon diodes, but in contrast to all earlier studies in cryogenic traps and rings where the rotational temperatures were always much higher than those of the storage devices at their lowest temperatures. Furthermore, we actively modify the rotational distribution through selective photodetachment to produce an OH$^-$ beam where 99.1$\pm$0.1 % of approximately one million stored ions are in the $J$=0 rotational ground state.
