  The process of a positron $\unicode{x2013}$ bound-electron annihilation with
simultaneous emission of two photons is investigated theoretically. A fully
relativistic formalism based on $\textit{ab initio}$ QED description of the
process is worked out. The developed approach is applied to evaluate the
annihilation of a positron with $K$-shell electrons of a silver atom, for which
a strong contradiction between theory and experiment was previously stated. The
results obtained here resolve this long-standing disagreement and, moreover,
demonstrate a sizeable difference with approaches so far used for calculations
of the positron $\unicode{x2013}$ bound-electron annihilation process, namely,
the $\textit{Lee's}$ and $\textit{impulse}$ approximations.
