We present a merged-beams study of reactions between HD^{+} ions, stored in the Cryogenic Storage Ring (CSR), and laser-produced ground-term C atoms. The molecular ions are stored for up to 20 s in the extreme vacuum of the CSR, where they have time to relax radiatively until they reach their vibrational ground state (within 0.5 s of storage) and rotational states with J≤3 (after 5 s). We combine our experimental studies with quasiclassical trajectory calculations based on two reactive potential energy surfaces. In contrast to previous studies with internally excited H_{2}^{+} and D_{2}^{+} ions, our results reveal a pronounced isotope effect, favoring the production of CH^{+} over CD^{+} across all collision energies, and a significant increase in the absolute rate coefficient of the reaction. Our experimental results agree well with our theoretical calculations for vibrationally relaxed HD^{+} ions in their lowest rotational states.