Abstract : We report an original procedure for theelaboration of very high-quality single crystals of superconductingHgBa2Ca2Cu3O8+et948; mercury cuprates. Thesesingle crystals are unique, with very high-quality surfacepaving the way for spectroscopic, transport, andthermodynamic probes in order to understand the holedopedcuprate phase diagram. Annealing allows one tooptimize Tc up to Tcmax = 133 K. The superconductivitytransition width of about 2 K indicates that they arehomogeneous. We show for the first time that, withadequate heat treatment, Hg-1223 can be largely underdopedand its doping level controlled. Importantly, thecrystal structure was studied in detail by single-crystal Xraydiffraction, and we have identified the signature of theunderdoping by a detailed sample characterization andmicro-Raman spectroscopy measurements.