  We report the quantitative experimental observation of the weak inertial-wave
turbulence regime of rotating turbulence. We produce a statistically steady
homogeneous turbulent flow that consists of nonlinearly interacting inertial
waves, using rough top and bottom boundaries to prevent the emergence of a
geostrophic flow. As the forcing amplitude increases, the temporal spectrum
evolves from a discrete set of peaks to a continuous spectrum. Maps of the
bicoherence of the velocity field confirm such a gradual transition between
discrete wave interactions at weak forcing amplitude, and the regime described
by weak turbulence theory (WTT) for stronger forcing. In the former regime, the
bicoherence maps display a near-zero background level, together with sharp
localized peaks associated with discrete resonances. By contrast, in the latter
regime the bicoherence is a smooth function that takes values of the order of
the Rossby number, in line with the infinite-domain and random-phase
assumptions of WTT. The spatial spectra then display a power-law behavior, both
the spectral exponent and the spectral level being accurately predicted by WTT
at high Reynolds number and low Rossby number.
