Antiferroelectric phase III of lead aluminium fluoride, PbsAI3Fw, Mr = 1477.9, tetragonal space group P4/n. At T = 295 K, a = 20.1738 (4) and c = 7.2205 (1) A,, V = 2939 (1) A 3, Z = 8, O m = 6.66 (5), Dx = 6.681 Mg m -3. For A(Mo Ka) = 0.71069 A,, /x = 58.0 mm-~, F(000) = 4960. The structure was determined from 18502 (1276 independent) Fm 2> -3~r(F~) with (sin 0)/,~ -< 0.703 A,-~ Least-squares refinement on wF~ resulted in R(Fm)= 0.0579 with Rint(Fm) = 0.048. PbsA13F19 undergoes a first-order phase transition from antiferroelectric to ferroelectric at about 110 K, with a wide thermal hysteresis.Transformation of the atomic coordinates of ferroelectric PbsCr3Fi9 previously measured at 295 K and comparison with those of antiferroelectric Pb5AI3FI9 at 295 K reveal differences between related atomic positions that range from 0.10 to 1.56 A. The origin of the first-order transition from the antiferroelectric phase III to ferroelectric phase IV in PbsAI3FI9 is shown to be associated with the orientational change from an eclipsed arrangement of A1F6 octahedra along the inversion and rotation-tetrad axes in phase III to a staggered arrangement along the rotationtetrad axes in phase IV.