New phases of the type M1-xBixO2-x/2 with a defect fluorite structure have been prepared where M is Zr or Hf. These apparently metastable phases were prepared by precipitation from solution followed by calcination at 600 degrees C. For the Hf1-xBixO2-x/2 system, x ranges from x = 0.40 to x = 0.75. For the Zr1-xBixO2-x/2 system, x ranges from x = 0.50 to x = 0.75. Heating the defect fluorite Hf1-xBixO2-x/2 phases to temperatures of 700 to 950 degrees C leads to the formation of Bi2Hf2O7 with the Bi2Sn2O7 structure. Thus, Bi2Hf2O7 is pseudocubic, but apparently face centered tetragonal with a = 21.66 Angstrom and c = 21.84 Angstrom. When Bi2Hf2O7 is heated to 1000 degrees C, it decomposes into HfO2 and Bi1.94Hf0.06O3.03 with the beta-Bi2O3 structure. Heating the defect fluorite Zr1-xBixO2-x/2 phases to higher temperatures never produced a Bi2Zr2O7 phase with the Bi2Sn2O7 structure. Instead, the compositional range of the defect fluorite structure steadily decreases with increasing temperature, and this phase had completely decomposed at 750 degrees C into ZrO2 and Bi1.84Zr0.16O3.08 with the beta-Bi2O3 structure.