We demonstrate the use of dual-comb spectroscopy for isotope ratio measurements. We show that the analysis spectral range of a free-running near-infrared dual-comb spectrometer can be extended to the midinfrared by difference frequency generation to target specific spectral regions suitable for such measurements and especially the relative isotopic ratio $\delta {}^{13}\mathrm{C}$. The measurements performed present very good repeatability over several days with a standard deviation below 2‰ for a recording time of a few tens of seconds, and the results are compatible with measurements obtained using an isotope ratio mass spectrometer. Our setup also shows the possibility to target several chemical species without any major modification, which can be used to measure other isotopic ratios. Further improvements could decrease the uncertainties of the measurements, and the spectrometer could thus compete with isotope ratio spectrometers currently available on the market.