We compute the relic abundance of the right-handed sneutrinos in the supersymmetric F_D-term model of hybrid inflation. As well as providing a natural solution to the μ- and gravitino overabundance problems, the F_D-term model offers a new viable candidate to account for the cold dark matter in the Universe: the lightest right-handed sneutrino. In particular, the F_D-term model predicts a new quartic coupling of purely right-handed sneutrinos to the Higgs doublets that thermalizes the sneutrinos and makes them annihilate sufficiently fast to a level compatible with the current cosmic microwave background data. We analyze this scenario in detail and identify favourable regions of the parameter space within the framework of minimal supergravity, for which the lightest right-handed sneutrino becomes the thermal dark matter, in agreement with WMAP observations of cosmological inflation. Constraints derived from direct dark matter searches experiments are presented.