Zurek suggested (Nature 317, 505; 1985) that the Kibble mechanism, through which topological defects such as cosmic strings are believed to have been created in the early Universe, can also result in the formation of topological defects in liquid 4He, i.e. quantised vortices, during rapid quenches through the superfluid transition. Preliminary experiments (Hendry et al., Nature 368, 315; 1994) seemed to support this idea in that the quenches produced the predicted high vortex-densities. The present paper describes a new experiment incorporating a redesigned expansion cell that minimises vortex creation arising from conventional hydrodynamic flow. The post-quench line-densities of vorticity produced by the new cell are no more than 1010 m-2, a value that is at least two orders of magnitude less than the theoretical prediction. We conclude that most of the vortices detected in the original experiment must have been created through conventional flow processes.