  We report an original procedure for the elaboration of very high quality
single crystals of superconducting HgBa2Ca2Cu3O8+{\delta} 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. The superconductivity transition width of
about 2 K indicates they are homogeneous. Annealing allows to optimize Tc up to
Tc$^{max}$ = 133 K. We show for the first time that with adequate heat
treatment, Hg-1223 can be largely under-doped 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 under-doping by
a detailed sample characterization and micro-Raman spectroscopy measurements.
