We present ALMA Cycle 4 observations of CO(1-0), CO(3-2), and $^{13}$CO(3-2) line emission in the brightest cluster galaxy of RXJ0821+0752. This is one of the first detections of $^{13}$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 $8~{rm kpc}$ in length. The detected $^{13}$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 $^{12}$CO spectra. The narrower component ($60~{rm km}~{rm s}^{-1}$ FWHM) is consistent in both velocity centroid and linewidth with $^{13}$CO(3-2) emission, while the broader ($130-160~{rm km}~{rm s}^{-1}$), slightly blueshifted wing has no associated $^{13}$CO(3-2) emission. A simple local thermodynamic model indicates that the $^{13}$CO emission traces $2.1times 10^{9}~{rm M}_odot$ of molecular gas. Isolating the $^{12}$CO velocity component that accompanies the $^{13}$CO emission yields a CO-to-H$_2$ conversion factor of $alpha_{rm CO}=2.3~{rm M}_{odot}~({rm K~km~s^{-1}})^{-1}$, which is a factor of two lower than the Galactic value. Adopting the Galactic CO-to-H$_2$ conversion factor in brightest cluster galaxies 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.
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