While mixtures of Gaussian distributions have been studied for more than a century (Pearson, 1894), the construction of a reference Bayesian analysis of those models still remains unsolved, with a general prohibition of the usage of improper priors (Fr\"uwirth-Schnatter, 2006) due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution (Richardson and Green, 1997). By creating a new parameterisation cantered on the mean and variance of the mixture distribution itself, we are able to develop here a genuine non-informative prior for Gaussian mixtures with an arbitrary number of components. We demonstrate that the posterior distribution associated with this prior is almost surely proper and provide MCMC implementations that exhibit the expected exchangeability. While we only study here the Gaussian case, extension to other classes of location-scale mixtures is straightforward.