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von Neumann's original proof is beautiful and quite simple. It deduces the Stone-von Neumann theorem from the Plancherel theorem. This proof can be found in Folland's book "Harmonic analysis in phase space". I have also tried to give an exposition of this proof, along with a proof of a generalization of this theorem due to Mackey, in my paper "An Easy Proof of the Stone-von Neumann-Mackey Theorem" (Expositiones Mathematicae, 24(1):110-118, 2011, also available on the arXiv).