Privacy and trust affect our strategic thinking, yet they have not been precisely modeled in mechanism design. In settings of incomplete information, traditional implementations of a normal-form mechanism ---by disregarding the players' privacy, or assuming trust in a mediator--- may not be realistic and fail to reach the mechanism's objectives. We thus investigate implementations of a new type.We put forward the notion of a perfect implementation of a normal-form mechanism M: in essence, an extensive-form mechanism exactly preserving all strategic properties of M, WITHOUT relying on a trusted mediator or violating the privacy of the players. We prove that ANY normal-form mechanism can be perfectly implemented by a PUBLIC mediator using envelopes and an envelope-randomizing device (i.e., the same tools used for running fair lotteries or tallying secret votes). Differently from a trusted mediator, a public one only performs prescribed public actions, so that everyone can verify that he is acting properly, and never learns any information that should remain private.