Precision measurements, dark matter direct detection and LHC Higgs searches in a constrained NMSSM

22 pages, 8 figures. Note added and references addedInternational audienceWe reexamine the constrained version of the Next-to-Minimal Supersymmetric Standard Model with semi universal parameters at the GUT scale (CNMSSM). We include constraints from collider searches for Higgs and SUSY particles, upper bound on the relic density of dark matter, measurements of the muon anomalous magnetic moment and of B-physics observables as well as direct searches for dark matter. We then study the prospects for direct detection of dark matter in large scale detectors and comment on the prospects for discovery of heavy Higgs states at the LHC