TOPOLOGICAL defects formed during a rapid symmetry-breaking phase transition in the early Universe(1,2) could be responsible for seeding large-scale structure, for the anisotropy of the microwave background radiation, and for the predominance of matter over antimatter(3,4). The theory describing this cosmological phase transition is formally analogous to that describing the transition to the superfluid state in liquid He-3, so that in principle the process of cosmological defect formation can be modelled in the laboratory. Here we report the results of an experiment in which the ’primordial fireball’ is mimicked using a neutron-induced nuclear reaction (n + He-3 --> p + He-3 + 0.76 MeV) to heat small regions of superfluid He-3 above the superfluid transition temperature. These bubbles of normal liquid cool extremely rapidly, and we find that their transition back to the superfluid state is accompanied the formation of a random network of vortices (the superfluid analogue of cosmic strings). We monitor the evolution of this defect state by rotating the superfluid sample, allowing vortices to escape from the network and thus be probed individually. Our results provide clear confirmation of the idea that topological defects form at a rapid second-order phase transition, and give quantitative support to the Kibble-Zurek mechanism(5,6) of cosmological defect formation.