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The CDMS II experiment has observed three events which may have arisen from weakly interacting massive particle (WIMP) dark matter (DM) with mass of order 9 GeV colliding with nuclei. Although the implied WIMP parameter region seems to be excluded by limits from the XENON experiments, it is interesting that most of this tension can go away if the WIMP-nucleon interaction violates isospin. This motivates us to explore some of the implications for models in which a real gauge-singlet scalar particle, the darkon, serves as the WIMP, taking into account the recent discovery of a Higgs boson at the LHC and Planck determination of the DM relic density. In the simplest scenario, involving only the standard model plus a darkon, the Higgs boson is largely invisible due to its decay into a pair of darkons having the WIMP mass suggested by CDMS II and hence cannot be identified with the one found at the LHC. We find, on the other hand, that a two-Higgs-doublet model supplemented with a darkon has ample parameter space to accommodate well both the new potential DM hint from CDMS II and the Higgs data from the LHC, whether or not the darkon-nucleon interaction conserves isospin.