The effect of long-term high-temperature annealing on the phase composition, local crystal structure, and oxygen-ion conductivity of SOFC membranes based on zirconium dioxide solid solutions was studied. Crystals with the composition of (ZrO2)0.99−x(Sc2O3)x(R2O3)0.01 (where x = 0.08–0.1; R-Yb, Y, Tb, Gd) were obtained by the method of directed melt crystallization in a cold crucible. The crystals were annealed in air at a temperature of 1000 °C for 400 h. The phase analysis of the crystals before and after annealing was studied by X-ray diffractometry and Raman spectroscopy. The study of the ionic conductivity of the crystals was carried out by the method of impedance spectroscopy in the temperature range 400–900 °C. It has been shown that when various rare earth cations (Yb, Y, Tb, and Gd) are used, the maximum conductivity is observed for the compositions (ZrO2)0.91(Sc2O3)0.08(Yb2O3)0.01, (ZrO2)0.89(Sc2O3)0.1(Y2O3)0.01, (ZrO2)0.90(Sc2O3)0.09(Tb2O3)0.01, and (ZrO2)0.89(Sc2O3)0.1(Gd2O3)0.01. At the same time, these crystals have a highly symmetrical pseudocubic structure, which is retained even after crystal annealing. At comparable concentrations of Sc2O3, the conductivity of crystals decreases with an increase in the ionic radius of the rare earth cation. The high-temperature degradation of the conductivity is also discussed depending on the type of rare earth oxide and the concentration of scandium oxide.