  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.
