  We perform highly accurate calculations of the leading order QED correction
to the ground electronic state of molecular hydrogen. Numerical results are
obtained for a grid of the internuclear distances $R=0-10$ au with the relative
precision of about $10^{-8}$. The major numerical uncertainty of previous QED
results [K. Piszczatowski et al., JCTC $\textbf{5}$, 3039 (2009)] has been
eliminated. Nevertheless, the discrepancy with measurements in HD at the level
of 1.9 $\sigma$ persists.
