Synchrotron radiation (SR) has proven to be an invaluable contributor to the field of molecular spectroscopy, particularly in the terahertz region (1-10 THz) where its bright and broadband properties are currently unmatched by laboratory sources. However, measurements using SR are currently limited to a resolution of around 30 MHz, due to the limits of Fourier-transform infrared spectroscopy. To push the resolution limit further, we have developed a spectrometer based on heterodyne mixing of SR with a newly available THz molecular laser, which can operate at frequencies ranging from 1 to 5.5 THz. This spectrometer can record at a resolution of 80 kHz, with 5 GHz of bandwidth around each molecular laser frequency, making it the first SR-based instrument capable of sub-MHz, Doppler-limited spectroscopy across this wide range. This allows closely spaced spectral features, such as the effects of internal dynamics and fine angular momentum couplings, to be observed. Furthermore, mixing of the molecular laser with a THz comb is demonstrated, which will enable extremely precise determinations of molecular transition frequencies.