The nature of the transformation by which a supercooled liquid 'freezes' to a glass-the glass transition-is a central issue in condensed matter physics but also affects many other fields, including biology. Substantial progress has been made in understanding this phenomenon over the past two decades, yet many key questions remain. In particular, the factors that control the temperature-dependent relaxation and viscous properties of the liquid phase as the glass transition is approached (that is, whether the glass-forming liquid is 'fragile' or 'strong') remain unclear. Here we show that the fragility of a glass-forming liquid is intimately linked to a very basic property of the corresponding glass phase: the relative strength of shear and bulk moduli, or Poisson's ratio.