In this paper, Comon’s conventional identifiability theorem for Independent Component Analysis (ICA) is extended to the case of mixtures where several gaussian sources are present. We show, in an original and constructive proof, that using the conventional mutual information minimization framework, the separation of all the non- gaussian sources is always achievable (up to scaling factors and permutations). In particular, we prove that a suitably designed optimization framework is capable of seamlessly handling both the case of one single gaussian source being present in the mixture (separation of all sources achievable), as well as the case of multiple gaussian signals being mixed together with non-gaussian signals (only the non-gaussian sources can be extracted).