SPHERES can be stacked into two close-packed crystalline arrangements, face-centred cubic (f.c.c.) and hexagonal closed-packed (h.c.p.), which have identical close-packed volumes’ and very similar equations of state(2). But they are structurally distinct, implying that they might have different thermodynamic properties and stabilities. Finding a difference in free energy between the two structures has been the objective of much theoretical(3,4) and computational(5-7) effort, but without a conclusive resolution, Here I report that a significant difference in the pressure-volume (P-V) behaviour can be detected in the vicinity of a mechanical instability point within a single-occupancy cell model(8) of these packings, This model provides an exact thermodynamically reversible path between the two structures, and sa, the P-V isotherms can be integrated to obtain the Gibbs free-energy difference. I find that the f.c.c. phase is more stable by around 0.005RT per mol (where R is the universal gas constant); as the enthalpy difference is negligible, this implies that the entropy difference is of the order of 0.005R for all temperatures up to the melting point.