The Institute for Astrophysics and Geophysics solar observatory is producing high-fidelity, ultra-high-resolution spectra (R > 500000) of the spatially resolved surface of the Sun using a Fourier transform spectrometer (FTS). The radial velocity (RV) calibration of these spectra is currently performed using absorption lines from Earth’s atmosphere, limiting the precision and accuracy. To improve the frequency calibration precision and accuracy, we use a Fabry–Pérot etalon (FP) setup that is an evolution of the CARMENES FP design and an iodine cell in combination. To create an accurate wavelength solution, the iodine cell is measured in parallel with the FP. The FP is then used to transfer the accurate wavelength solution provided by the iodine via a simultaneous calibration of solar observations. To verify the stability and precision of the FTS, we perform parallel measurements of the FP and an iodine cell. The measurements show an intrinsic stability of the FTS of a level of 1 m s ^{− 1} over 90 h. The difference between the FP RVs and the iodine cell RVs show no significant trends during the same time span. The root mean square of the RV difference between the FP and iodine cell is 10.7 cm s ^{− 1}, which can be largely attributed to the intrinsic RV precisions of the iodine cell and the FP (10.2 and 1.0 cm s ^{− 1}, respectively). This shows that we can calibrate the FTS to a level of 10 cm s ^{− 1}, competitive with current state-of-the-art precision RV instruments. Based on these results, we argue that the spectrum of iodine can be used as an absolute reference to reach an RV accuracy of 10 cm s ^{− 1}.