  The two-photon annihilation of a positron with an electron bound in the 1s
state of a H-like ion is calculated within the fully relativistic QED
framework. The interaction with the nucleus is treated nonperturbatively, thus
allowing the calculations to be carried out for the annihilation with
strongly-bound inner shells of heavy ions. Infrared divergences, appearing when
one of the emitted photons approaches the low-frequency limit, are accurately
eliminated from final expressions. The total cross section of the two-photon
and one-photon annihilation processes are compared for a wide range of
collision energies and nuclear charge numbers. It is demonstrated that the
two-photon annihilation channel dominates over the one-photon channel for the
low and medium-Z ions, whereas for the high-Z ions the situation reverses.
