The standard model ( SM ) plus a real gauge-singlet scalar field dubbed darkon ( SM+D ) is the simplest model possessing a weakly interacting massive particle ( WIMP ) dark-matter candidate . The upper limits for the WIMP-nucleon elastic cross-section as a function of WIMP mass from the recent XENON10 and CDMS II experiments rule out darkon mass ranges from 10 to ( 50 , 70 , 75 ) GeV for Higgs-boson masses of ( 120 , 200 , 350 ) GeV , respectively . This may exclude the possibility of the darkon providing an explanation for the gamma-ray excess observed in the EGRET data . We show that by extending the SM+D to a two-Higgs-doublet model plus a darkon the experimental constraints on the WIMP-nucleon interactions can be circumvented due to suppression occurring at some values of the product \tan \alpha \tan \beta , with \alpha being the neutral-Higgs mixing angle and \tan \beta the ratio of vacuum expectation values of the Higgs doublets . We also comment on the implication of the darkon model for Higgs searches at the LHC .