[en] Highlights: • Lamb-dips of 45 lines in the $v_1+3v_3$ band of ¹²C₂H₂ in the 786–793 nm region were recorded by cavity ring-down saturation spectroscopy. • Line positions were determined with an accuracy as good as 3.2 kHz (relatively $8\times {10}^{-12}$). The ground state combination differences obtained in this work agree with the most precisely reported results from the $v_1+v_3$ band near $1.5\mu$ m. • Upper energies of these transitions have an average shift of $0.01{\text{cm}}^{-1}$ compared to those recently derived from the MARVEL method. Lamb-dips of the ro-vibrational lines of ¹²C₂H₂ near 789 nm were recorded using cavity ring-down saturation spectroscopy. Calibrated by an optical frequency comb, frequencies of 45 acetylene lines were determined with an accuracy of $1.1\times {10}^{-7}$ cm${}^{-1}$ ($\delta \nu /\nu =8\times {10}^{-12}$), which is over two orders of magnitude more accurate than previous Doppler-limited studies. An averaged shift of about 0.01 cm${}^{-1}$ were found by comparing the upper energies obtained in this work to those recently presented by Chubb et al. from a MARVEL analysis.