Phase transitions are dramatic phenomena: water freezes into ice, atomic spins spontaneously align in a magnet, and liquid helium becomes superfluid. Sometimes, such a drastic change in behaviour is accompanied by a visible change in appearance. The hallmark of Bose-Einstein condensation and superfluidity in trapped, weakly interacting Bose gases is the sudden formation of a dense central core inside a thermal cloud. However, in strongly interacting gases--such as the recently observed fermionic superfluids--there is no longer a clear separation between the superfluid and the normal parts of the cloud. The detection of fermion pair condensates has required magnetic field sweeps into the weakly interacting regime, and the quantitative description of these sweeps presents a major theoretical challenge. Here we report the direct observation of the superfluid phase transition in a strongly interacting gas of 6Li fermions, through sudden changes in the shape of the clouds--in complete analogy to the case of weakly interacting Bose gases. By preparing unequal mixtures of the two spin components involved in the pairing, we greatly enhance the contrast between the superfluid core and the normal component. Furthermore, the distribution of non-interacting excess atoms serves as a direct and reliable thermometer. Even in the normal state, strong interactions significantly deform the density profile of the majority spin component. We show that it is these interactions that drive the normal-to-superfluid transition at the critical population imbalance of 70 +/- 5 per cent (ref. 12).