By reducing both the internal and translational temperature of any species down to a few kelvins, the buffer-gas-cooling (BGC) technique has the potential to dramatically improve the quality of ro-vibrational molecular spectra, thus offering unique opportunities for transition frequency measurements with unprecedented accuracy. However, the difficulty in integrating metrological-grade spectroscopic tools into bulky cryogenic equipment has hitherto prevented from approaching the kHz level even in the best cases. Here, we overcome this drawback by an original opto-mechanical scheme which, effectively coupling a Lamb-dip saturated-absorption cavity ring-down spectrometer to a BGC source, allows us to determine the absolute frequency of the acetylene (ν1+ν3) R(1)e transition at 6561.0941 cm-1 with a fractional uncertainty as low as 6∙10-12. By improving the previous record with buffer-gas-cooled molecules by one order of magnitude, our approach paves the way for a number of ultra-precise low-temperature spectroscopic studies, aimed at both fundamental Physics tests and optimized laser cooling strategies.