Negative kinetic energies correspond to ghost degrees of freedom, which are potentially of relevance for cosmology, quantum gravity, and high energy physics. We present a novel wide class of stable mechanical systems where a positive energy degree of freedom interacts with a ghost. These theories have Hamiltonians unbounded from above and from below, are integrable, and contain free functions. We show analytically that their classical motion is bounded for all initial data. Moreover, we derive conditions allowing for Lyapunov stable equilibrium points. A subclass of these stable systems has simple polynomial potentials with stable equilibrium points entirely due to interactions with the ghost. All these findings are fully supported by numerical computations which we also use to gather evidence for stability in various nonintegrable systems.