The Al-Mg phase diagram has been reinvestigated in the vicinity of the stability range of the Samson phase, β-Mg2Al3 (cF1168). For the composition Mg38.5Al61.5, this cubic phase, space group Fd3̄m (no 227), a = 28.242(1) Å, V = 22526(2) Å3, undergoes at 214°C a first-order phase transition to rhombohedral β′-Mg2Al3, (hR293), a = 19.968(1) Å, c = 48.9114(8) Å, V = 16889(2) Å3, (i.e. 22519 Å3 for the equivalent cubic unit cell) space group R3m (no 160), a subgroup of index four of Fd3̄m. The structure of the β-phase has been redetermined at ambient temperature as well as in situ at 400°C. It essentially agrees with Samson's model, even in most of the many partially occupied and split positions. The structure of β′-Mg 2Al3 is closely related to that of the β-phase. Its atomic sites can be derived from those of the β-phase by group-theoretical considerations. The main difference between the two structures is that all atomic sites are fully occupied in case of the β′-phase. The reciprocal space, Bragg as well as diffuse scattering, has been explored as function of temperature and the β- to β′-phase transition was studied in detail. The microstructures of both phases have been analyzed by electron microscopy and X-ray topography showing them highly defective. Finally, the thermal expansion coefficients and elastic parameters have been determined. Their values are somewhere in between those of Al and Mg.