Measurements of the gas mass are necessary to determine the planet formation potential of protoplanetary disks. Observations of rare CO isotopologues are typically used to determine disk gas masses; however, if the line emission is optically thick this will result in an underestimated disk mass. With the Atacama Large Millimeter/submillimeter Array we have detected the rarest stable CO isotopologue, ¹³C¹⁷O, in a protoplanetary disk for the first time. We compare our observations with the existing detections of ¹²CO, ¹³CO, C¹⁸O, and C¹⁷O in the HD 163296 disk. Radiative transfer modeling using a previously benchmarked model, and assuming interstellar isotopic abundances, significantly underestimates the integrated intensity of the ¹³C¹⁷O J = 3–2 line. Reconciliation between the observations and the model requires a global increase in CO gas mass by a factor of 3.5. This is a factor of 2–6 larger than previous gas mass estimates using C¹⁸O. We find that C¹⁸O emission is optically thick within the snow line, while the ¹³C¹⁷O emission is optically thin and is thus a robust tracer of the bulk disk CO gas mass.