Taking into account the available accelerator and astrophysical constraints, the mass of the lightest neutral Higgs boson h in the minimal supersymmetric extension of the Standard Model with universal soft supersymmetry-breaking masses (CMSSM) has been estimated to lie between 114 and ~130 GeV. Recent data from ATLAS and CMS hint that mh~125 GeV, though mh~119 GeV may still be a possibility. Here we study the consequences for the parameters of the CMSSM and direct dark matter detection if the Higgs hint is confirmed, focusing on the strips in the (m1/2,m0) planes for different tanβ and A0 where the relic density of the lightest neutralino χ falls within the range of the cosmological cold dark matter density allowed by WMAP and other experiments. We find that if mh~125 GeV focus-point strips would be disfavoured, as would the low-tanβ τ̃-χ and t̃1-χ coannihilation strips, whereas the τ̃-χ coannihilation strip at large tanβ and A0>0 would be favoured, together with its extension to a funnel where rapid annihilation via direct-channel H/A poles dominates. On the other hand, if mh~119 GeV more options would be open. We give parameterisations of WMAP strips with large tanβ and fixed A0/m0>0 that include portions compatible with mh=125 GeV, and present predictions for spin-independent elastic dark matter scattering along these strips. These are generally low for models compatible with mh=125 GeV, whereas the XENON100 experiment already excludes some portions of strips where mh is smaller.