Precision spectroscopy of molecular hydrogen can be used to test the quantum electrodynamics theory and determine the proton-to-electron mass ratio in a four-body system. Saturated absorption spectroscopy of the R(1) line in the (2-0) band of HD is measured at 76 K. An asymmetric line profile is observed and analyzed, and the transition frequency is predicted to be $217\phantom{\rule{0.16em}{0ex}}105\phantom{\rule{0.16em}{0ex}}182\phantom{\rule{0.16em}{0ex}}284\phantom{\rule{0.16em}{0ex}}{(11)}_{\mathrm{stat}}{(27)}_{\mathrm{sys}}$ kHz. The value is nine times more accurate than our previous result obtained at room temperature and differs by $6.6\ensuremath{\sigma}$ from the recent result derived with a different line-shape model.