An undominated Nash equilibrium is a Nash equilibrium in which no agent selects a weakly dominated strategy. A mechanism is bounded if every dominated strategy is dominated by an undominated strategy. The authors give necessary and sufficient conditions for an implementation in undominated Nash equilibrium when mechanisms are bounded, i.e., characterize the social choice correspondences which are implementable in such an equilibrium under such a mechanism. Many social correspondences of interest are shown to be implemented by bounded mechanisms. For bounded mechanisms admitting any mixed strategy equilibrium for any von Neumann-Morgenstern utility functions, the authors investigate implementation in economic settings.