Many liquids cooled to low temperatures form glasses (amorphous solids) instead of crystals. As the glass transition is approached, molecules become localized and relaxation times increase by many orders of magnitude(1). Many features of this ＇slowing down＇ are reasonably well described(2) by the mode-coupling theory of supercooled liquids(3). The ideal form of this theory predicts a dynamical critical temperature T-c at which the molecules become permanently trapped in the ＇cage＇ formed by their neighbours, and vitrification occurs. Although there is no sharp transition, because molecules do eventually escape their cage, its signature can still be observed in real and simulated liquids. Unlike conventional critical phenomena (such as the behaviour at the liquid-gas critical point), the mode-coupling transition is not accompanied by a diverging static correlation length. But simulation(4-10) and experiment(11,12) show that liquids are dynamically heterogeneous, suggesting the possibility of a relevant ＇dynamical＇ length scale characterizing the glass transition. Here we use computer simulations to investigate a melt of short, unentangled polymer chains over a range of temperatures for which the mode-coupling theory remains valid. We find that although density fluctuations remain short-ranged, spatial correlations between monomer displacements become long-ranged as T-c is approached on cooling. In this way, we identify a growing dynamical correlation length, and a corresponding order parameter, associated with the glass transition. This finding suggests a possible connection between well established concepts in critical phenomena and the dynamics of glass-forming liquids.