The $F^{+}$ center consisting of an electron trapped at an oxygen vacancy ($V_{\mathrm{O}}$) was investigated in oxygen deficient ${\mathrm{Y}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$ (YAG) garnet crystals by electron paramagnetic resonance (EPR) techniques. The measurements were performed in the wide temperature interval 5–450 K and the frequency range 9.4–350 GHz using both the conventional continuous-wave and pulse EPR experiments. Pulse electron-nuclear double resonance was applied to resolve the hyperfine interaction of the trapped electron with surrounding nuclei. The measurements show that at low temperatures, $T<50$ K, the EPR spectrum of the $F^{+}$ center is anisotropic with $g$ factors in the range 1.999–1.988 and originates from three magnetically inequivalent positions of the center in a garnet lattice. As the temperature increases, the EPR spectrum becomes isotropic suggesting a motional averaging of the anisotropy due to motion of the $F^{+}$-center electron between neighboring oxygen vacancies. With further increase of temperature up to $T$ > 200 K, we observed delocalization of the $F^{+}$-center electron into the conduction band with the activation energy about 0.4–0.5 eV that resulted in substantial narrowing of the EPR spectral line with simultaneous change of its shape from Gaussian to Lorentzian due to disappearance of the Fermi contact hyperfine field at ${}^{27}\mathrm{Al}$ and ${}^{89}\mathrm{Y}$ nuclei. Such temperature behavior of the $F^{+}$-center electron in YAG is completely similar to behavior of a donor electron in a semiconductor. Our findings are further supported by measurements of the conductivity and dielectric properties. In particular, these data show that the electrons in the conduction band are not homogeneously distributed in the crystal: There are highly conductive regions separated by poorly conductive dielectric layers. This leads to the so-called Maxwell-Wagner dielectric relaxation with huge apparent dielectric constant at low frequencies. This paper reports an observation of a donorlike behavior of an $F^{+}$ center in wide band-gap insulating crystals.