Although the birth of the Universe is inaccessible to experimental study, aspects of cosmological theories can nonetheless be explored in the laboratory. Tiny inhomogeneities in the mix of particles and radiation produced in the Big Bang grew into the clusters of galaxies that we see today, but how those inhomogeneities arose and grew is still unclear. Cosmologies based on grand unified theories suggest that a symmetry-breaking phase transition occurred via the Higgs mechanism about 10^{34} s after the Big Bang as the Universe cooled through a critical temperature of 10^{27}K. It has been proposed by Kibble that this transition may have generated defects in the geometry of space-time (such as cosmic strings), which provided the inhomogeneities on which galaxies subsequently condensed. Zurek has suggested that it might be possible to model this cosmological phase transition by a laboratory analogue, the superfluid transition of liquid 4He induced by fast adiabatic expansion through the critical density. Here we report the results of such an experiment. We observe copious production of quantized vortices, the superfluid analogue of cosmic strings. These results support Kibble's contention that such defects were available in the early Universe to seed galaxy formation.