The iron-rich intermetallics ${R}_{2}$Fe${}_{17}$ (where $R$ is a rare-earth element) proved unamenable to anisotropy control via interstitial doping. There is only one precedent, Sm${}_{2}$Fe${}_{17}$N${}_{3\ensuremath{-}\ensuremath{\delta}}$ (Sm${}_{2}$Fe${}_{17}$C${}_{3\ensuremath{-}\ensuremath{\delta}}$), where interstitial modification has stabilized an easy-axis anisotropy at all temperatures up to ${T}_{\mathrm{C}}$. All previous attempts to prepare a usable easy-axis ${R}_{2}$Fe${}_{17}$ hydride have failed. Now we have succeeded in preparing a high-quality single-phase Tb${}_{2}$Fe${}_{17}$H${}_{3}$ single crystal, which has the required easy-axis anisotropy between 0 K and ${T}_{\mathrm{C}}$ $=$ 560 K. At $T$ $=$ 300 K, Tb${}_{2}$Fe${}_{17}$H${}_{3}$ has the spontaneous magnetic moment ${M}_{\mathrm{s}}$ of 22.5 \ensuremath{\mu}${}_{\mathrm{B}}$ per formula unit and anisotropy field \ensuremath{\mu}${}_{0}$${H}_{\mathrm{a}}$ of 2.5 T. The main mechanism stabilizing the easy-axis anisotropy in hydrides is the same as in other similar compounds by way of boosting the leading crystal field parameter ${A}_{20}$. Terbium is rather special in having the Stevens factors such that ${\ensuremath{\alpha}}_{J}$ 0 and ${\ensuremath{\beta}}_{J}$ g 0, which is why the easy-axis anisotropy in Tb${}_{2}$Fe${}_{17}$ hydrides is also assisted by the fourth-order parameter ${A}_{40}$. This proves a decisive advantage over the compounds with $R$ $=$ Pr, Nd, Dy, or Ho where ${\ensuremath{\beta}}_{J}$ 0 and ${A}_{40}$ is a hindrance.