We report a high-precision calculation of the photodetachment energy of the hydrogen anion $H^{-}$, also known as the electron affinity of the hydrogen atom. The nonrelativistic bound-state energy is obtained using an exact three-body approach, and supplemented by leading relativistic, quantum-electrodynamic, finite-nuclear-size, and hyperfine corrections. Our result is $6083.06447(68){cm}^{-1}$ for the detachment to the hydrogen ground-state hyperfine level $(F=0)$, which is 220 times more precise than the best experimental determination to date, $6082.99(15){cm}^{-1}$, as reported by Lykke Beyond their intrinsic interest, these results provide critical input for antihydrogen physics, where controlled photodetachment of ${\bar{H}}^{+}$ offers a path to producing ultracold antihydrogen (and its isotopes) for precision experiments. We further determine the photodetachment thresholds for ${}^2{H}^{-}$ and ${}^3{H}^{-}$ into the ground hyperfine states of the corresponding hydrogenic atoms, yielding $6086.70679(68){cm}^{-1}$ for ${}^{2}H(F=1/2)$ and $6087.87924(68){cm}^{-1}$ for ${}^{3}H(F=0)$.