Cosmological constraints on the sum of the neutrino masses can be relaxed if the number density of active neutrinos is reduced compared to the standard scenario, while at the same time keeping the effective number of neutrino species Neff \approx 3 by introducing a new component of dark radiation. We discuss a UV complete model to realise this idea, which simultaneously provides neutrino masses via the seesaw mechanism. It is based on a U(1) symmetry in the dark sector, which can be either gauged or global. In addition to heavy seesaw neutrinos, we need to introduce \mathcal{O}(10) generations of massless sterile neutrinos providing the dark radiation. Then we can accommodate active neutrino masses with \sum mν \sim 1 eV, in the sensitivity range of the KATRIN experiment. We discuss the phenomenology of the model and identify the allowed parameter space. We argue that the gauged version of the model is preferred, and in this case the typical energy scale of the model is in the 10 MeV to few GeV range.