The process of a positron-bound-electron annihilation with simultaneous emission of two photons is investigated theoretically. A fully relativistic formalism based on an 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 longstanding disagreement and, moreover, demonstrate a sizable difference with approaches so far used for calculations of the positron-bound-electron annihilation process, namely, Lee's and the impulse approximations.