THEORIES of melting in two dimensions1-3 predict that it may have a very different character from the three-dimensional transition. In particular, Halperin and Nelson2 proposed that an intermediate hexatic phase, with long-range orientational but not positional order, might intervene between the two-dimensional solid and liquid. Hexatic order has since been identified in three-dimensional liquid-crystal phases4. Several liquid-crystal compounds that exhibit a hexatic mesophase can form free-standing films ranging from two to thousands of molecular layers in thick-ness, permitting experimental investigation of theories of melting in two dimensions, free from the effects of a substrate. We have previously studied transitions between liquid, hexatic and crystal-line phases in films of n-heptyl-4'-n-pentyloxybiphenyl-4-carboxylate (75OBC), and for films thicker than four layers we observed separate liquid-hexatic transitions for surface and inner layers5, revealed by heat-capacity measurements. Here we report that improvements to our measurement technique allow us to identify these transitions in individual molecular layers, even in films just three layers thick. We also observe a single liquid-hexatic transition in a two-layer film. These films therefore provide model systems for the study of truly two-dimensional phase transitions.