  We investigate one-photon annihilation of the electron-positron pair in the
field of a central, very heavy and positively charged atomic nucleus. The
explicit formula for the annihilation rate of this process $\Gamma^{(b)}_{1
\gamma}$ is derived. Our formula for this rate can directly be used to describe
the actual one-photon annihilation in the ground (bound) states of all
positronium hydrides HPs, quasi-stable triplet states of the positron-helium
atoms $e^{+}[$ He($2^{3}S_e$)] ions and other systems.
