We present ALMA Cycle 4 observations of CO(1-0), CO(3-2), and {sup 13}CO(3-2) line emission in the brightest cluster galaxy (BCG) of RXJ0821+0752. This is one of the first detections of {sup 13}CO line emission in a galaxy cluster. Half of the CO(3-2) line emission originates from two clumps of molecular gas that are spatially offset from the galactic center. These clumps are surrounded by diffuse emission that extends 8 kpc in length. The detected {sup 13}CO emission is confined entirely to the two bright clumps, with any emission outside of this region lying below our detection threshold. Two distinct velocity components with similar integrated fluxes are detected in the {sup 12}CO spectra. The narrower component (60 km s{sup −1} FWHM) is consistent in both velocity centroid and linewidth with {sup 13}CO(3-2) emission, while the broader (130–160 km s{sup −1}), slightly blueshifted wing has no associated {sup 13}CO(3-2) emission. A simple local thermodynamic model indicates that the {sup 13}CO emission traces 2.1 × 10{sup 9} M {sub ⊙} of molecular gas. Isolating the {sup 12}CO velocity component that accompanies the {sup 13}CO emission yields a CO-to-H{sub 2} conversion factor of α {sub CO} = 2.3 M {sub ⊙} (K km s{sup −1}){sup −1}, which is a factor of two lower than the Galactic value. Adopting the Galactic CO-to-H{sub 2} conversion factor in BCGs may therefore overestimate their molecular gas masses by a factor of two. This is within the object-to-object scatter from extragalactic sources, so calibrations in a larger sample of clusters are necessary in order to confirm a sub-Galactic conversion factor.