Dark matter (DM) is currently searched for with a variety of detection strategies. Accelerator searches are particularly promising, but even if weakly interacting massive particles are found at the Large Hadron Collider (LHC), it will be difficult to prove that they constitute the bulk of the DM in the Universe ΩDM. We show that a significantly better reconstruction of the DM properties can be obtained with a combined analysis of LHC and direct detection data, by making a simple Ansatz on the weakly interacting massive particles local density ρχ˜10, i.e., by assuming that the local density scales with the cosmological relic abundance, (ρχ˜10/ρDM)=(Ωχ˜10/ΩDM). We demonstrate this method in an explicit example in the context of a 24-parameter supersymmetric model, with a neutralino lightest supersymmetric particle in the stau coannihilation region. Our results show that future ton-scale direct detection experiments will allow to break degeneracies in the supersymmetric parameter space and achieve a significantly better reconstruction of the neutralino composition and its relic density than with LHC data alone.