The two-(fixed) center Coulomb problem possesses in nonrelativistic mechanics, besides the "classical" constants of motion, an additional constant of motion associated with dynamical symmetry and separability of the problem. When the motion of the nuclei is taken into account the constant of motion is destroyed but reappears as an adiabatic invariant. We have identified this novel adiabatic invariant for the unrestricted three-body problem in the limit of two heavy particles and one light particle. Unlike all other known adiabatic invariants, it is globally conserved to order delta = m/M in the light- to heavy-mass ratio. The significance of this invariant for the semiclassical description of the chemical bond and for diatomic molecules beyond the Born-Oppenheimer approximation is discussed.