We present ALMA Cycle 4 observations of CO(1-0), CO(3-2), and 13CO(3-2) line emission in the brightest cluster galaxy (BCG) of RXJ0821+0752. This is one of the first detections of 13CO 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 13CO 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 12CO spectra. The narrower component (60 km s-1 FWHM) is consistent in both velocity centroid and linewidth with 13CO(3-2) emission, while the broader (130-160 km s-1), slightly blueshifted wing has no associated 13CO(3-2) emission. A simple local thermodynamic model indicates that the 13CO emission traces 2.1 ×109 M o of molecular gas. Isolating the 12CO velocity component that accompanies the 13CO emission yields a CO-to-H2 conversion factor of α CO = 2.3 M o (K km s-1)-1, which is a factor of two lower than the Galactic value. Adopting the Galactic CO-to-H2 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.