We study the formation of a supermassive black hole (SMBH) binary and the shrinking of the separation of the two holes to sub-parsec scales starting from a realistic major merger between two gas-rich spiral galaxies with mass comparable to our Milky Way. The simulations, carried out with the adaptive mesh refinement (AMR) code RAMSES, are capable of resolving separations as small as 0.1 pc. The collision of the two galaxies produces a gravoturbulent rotating nuclear disc with mass (˜109 M⊙) and size (˜60 pc) in excellent agreement with previous smoothed particle hydrodynamics simulations with particle splitting that used a similar set-up (Mayer et al. 2007) but were limited to separations of a few parsecs. The AMR results confirm that the two black holes sink rapidly as a result of dynamical friction on to the gaseous background, reaching a separation of 1 pc in less than 107 yr. We show that the dynamical friction wake is well resolved by our model and we find good agreement with analytical predictions of the drag force as a function of the Mach number. Below 1 pc, black hole pairing slows down significantly, as the relative velocity between the sinking SMBH becomes highly subsonic and the mass contained within their orbit falls below the mass of the binary itself, rendering dynamical friction ineffective. In this final stage, the black holes have not opened a gap as the gaseous background is highly pressurized in the centre. Non-axisymmetric gas torques do not arise to restart sinking in absence of efficient dynamical friction, at variance with previous calculations using idealized equilibrium nuclear disc models. We believe that the rather `hot' equation of state we used to model the multiphase turbulent interstellar medium in the nuclear region is playing an important role in preventing efficient SMBH sinking inside the central parsec. We conclude with a discussion of the way forward to address sinking in gaseous backgrounds at sub-parsec scales approaching the gravitational wave regime.