We report an original procedure for the elaboration of very high-quality single crystals of superconducting HgBa2Ca2Cu3O8+δ mercury cuprates. These single crystals are unique, with very high-quality surface paving the way for spectroscopic, transport, and thermodynamic probes in order to understand the hole-doped cuprate phase diagram. Annealing allows one to optimize Tc up to Tcmax = 133 K. The superconductivity transition width of about 2 K indicates that they are homogeneous. We show for the first time that, with adequate heat treatment, Hg-1223 can be largely underdoped and its doping level controlled. Importantly, the crystal structure was studied in detail by single-crystal X-ray diffraction, and we have identified the signature of the underdoping by a detailed sample characterization and micro-Raman spectroscopy measurements.