We determine the strong coupling constant ${\ensuremath{\alpha}}_{s}$ and its energy dependence from the ${p}_{T}$ dependence of the inclusive jet cross section in $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. The strong coupling constant is determined over the transverse momentum range $50<{p}_{T}<145\text{ }\text{ }\mathrm{GeV}$. Using perturbative QCD calculations to order $\mathcal{O}({\ensuremath{\alpha}}_{s}^{3})$ combined with $\mathcal{O}({\ensuremath{\alpha}}_{s}^{4})$ contributions from threshold corrections, we obtain ${\ensuremath{\alpha}}_{s}({M}_{Z})={0.1161}_{\ensuremath{-}0.0048}^{+0.0041}$. This is the most precise result obtained at a hadron-hadron collider.