Sodium niobate undergoes an antiferroelectric-ferroelectric transition (P ~ N) at temperatures below about -110°C. The ferroelectric low-temperature phase (N) is rhombohedral, with space group R3c.Though a perovskite structure, it is isostructural with LiNbO3. Using pseudocubic axes of reference, a=2 × 3"9083 (5) ~, ~= 89 ° 13 (1)' at -150°C. The NbO6 octahedra remain nearly regular, but are tilted about the triad axis; displacements of Nb atoms from the centres of octahedra are parallel to this axis. The temperature dependence of the lattice parameters of both phases, N and P, below room temperature is reported, and also that of the magnitude of the Nb displacement in the antiferroelectric phase P. Experimental details of the low-temperature work are given. Evidence used in the structure determination comes from the juxtaposition of high-angle reflexions from twin components in phase N, and the orientation of domains of phase N relative to the parent phase P. The transformation is more rapid and more complete if the parent crystal is single-domain; if it is twinned, both phases coexist for long periods down to the lowest temperatures studied. The relation of the structure to that of other niobates is discussed, and some general principles are put forward from which the occurrence of different structures and the transitions between them can be understood.