In this work we report the results of calculation for quantum-mechanical rotational transitions in molecular hydrogen, ${\mathrm{H}}_{2}$, induced by an ultracold ground-state antihydrogen atom ${\overline{\mathrm{H}}}_{1s}$. The calculations are accomplished using a nonreactive close-coupling quantum-mechanical approach. The ${\mathrm{H}}_{2}$ molecule is treated as a rigid rotor. The total elastic-scattering cross section ${\ensuremath{\sigma}}_{\mathrm{el}}(\ensuremath{\epsilon})$ at energy $\ensuremath{\epsilon}$, state-resolved rotational transition cross sections ${\ensuremath{\sigma}}_{{\mathit{jj}}^{'}}(\ensuremath{\epsilon})$ between states $j$ and ${j}^{'}$, and corresponding thermal rate coefficients ${k}_{{\mathit{jj}}^{'}}(T)$ are computed in the temperature range $0.004 \mathrm{K}\ensuremath{\lesssim}T\ensuremath{\lesssim}$ 4 K. Satisfactory agreement with other calculations (variational) has been obtained for ${\ensuremath{\sigma}}_{\mathrm{el}}(\ensuremath{\epsilon})$.
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