We consider the problem of reconstructing a linear combination of integer shifts of a so-called totally positive function of Gaussian type. In analogy to Beurling's sampling theorem for the Paley-Wiener space of entire functions, we prove that every separated set with lower Beurling density above 1 allows for such stable reconstruction. In view of the known necessary density conditions, this result is optimal. We also present applications to the characterization of spanning properties of Gabor systems (dictionaries of functions consisting of time-frequency shifts of a single generator). Joint work with Karlheinz Groechenig and Joachim Stoeckler.