We report on NMR studies of the quasi one--dimensional (1D) antiferromagnetic $S=1/2$ chain cuprate LiCuVO$_4$ in magnetic fields $H$ up to $mu_0H$ = 30 T ($approx 70$% of the saturation field $H_{rm sat}$). NMR spectra in fields higher than $H_{rm c2}$ ($mu_0H_{rm c2} approx 7.5$ T) and temperatures $T<T_{rm N}$ can be described within the model of a spin-modulated phase in which the magnetic moments are aligned parallel to the applied field $H$ and their values alternate sinusoidally along the magnetic chains. Based on theoretical concepts about magnetically frustrated 1D chains, the field dependence of the modulation strength of the magnetic structure is deduced from our experiments. Relaxation time $T_2$ measurements of the $^{51}$V nuclei show that $T_2$ depends on the particular position of the probing $^{51}$V nucleus with respect to the magnetic copper moments within the 1D chains: the largest $T_2$ value is observed for the vanadium nuclei which are very next to the magnetic Cu$^{2+}$ ion with largest ordered magnetic moment. This observation is in agreement with the expectation for the spin-modulated magnetic structure. The $(H,T)$ magnetic phase diagram of LiCuVO$_4$ is discussed.
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