The CoGeNT and DAMA/LIBRA experiments have found evidence for the spin-independent scattering from nuclei of a light dark matter (DM) particle, 7-12 GeV, which is not excluded by the XENON DM experiments. We show that this putative DM signal can be explained by a complex scalar singlet extension of the standard model (CSM), with a thermal cosmological DM density, and a Higgs sector that is consistent with LEP constraints. We make predictions for the masses, production, and decays of the two Higgs mass eigenstates and describe how the Higgs and DM particles can be discovered at the LHC.