Abstract : We present a Nuclear Magnetic Resonance (NMR) study of azurite, Cu3(CO3)2(OH)2, a quantum spin system. This compound has been recognised as a model system for a quasi-1D, frustrated, 'diamond' chain of S=1/2 spins beared by Cu2+ ions. In the magnetisation curve as a function of magnetic field it presents, between 11 and 30 T and at very low temperatures, a plateau at 1/3 of the saturation magnetisation. We performed Cu NMR measurements in azurite at T=1.5 K in order to determine its microscopic magnetic structure. The obtained results show that the 'dimer' of two more strongly coupled spins is approximately in a singlet state while the third spin (the 'monomer') is almost fully polarised. This confirms that the electronic configuration of the 1/3 plateau is a new quantum state without classical analogue [F. Aimo et al., Phys. Rev. Lett. 102, 127205, (2009)]. By very high magnetic field proton NMR, between 31 and 34 T and at T=0.6 K, we have also studied the transition region between the 1/3 plateau and the full polarisation of the system in order to test for the possible existence of a 2/3 plateau. This plateau is expected in rather exceptional case when longitudinal spin correlations are dominant and stabilise an incommensurable longitudinal order. However, our analysis showed that the symmetric splitting of NMR spectra corresponds to an antiferromagnetic transverse and not longitudinal order, which is incompatible with the existence of a 2/3 plateau.