In unimodular gravity the stress-energy tensor of matter is not necessarily conserved, and so the theory offers a natural framework for interacting dark energy models, where dark energy has a constant equation of state w = −1. We derive the equations of motion for linear cosmological perturbations in interacting dark energy models of this class, focusing on the scalar sector. Then, we consider a specific model where the energymomentum transfer potential is proportional to the energy density of cold dark matter; this transfer potential has the effect of inducing an effective equation of state w eff = 0 for cold dark matter. We analyze in detail the evolution of perturbations during radiation domination on super-Hubble scales, finding that the well-known large-scale instability that affects a large class of interacting dark energy models is absent in this model. To avoid a gradient instability, energy must flow from dark matter to dark energy. Finally, we show that interacting dark energy models with w = −1 are equivalent to a class of generalized dark matter models.