We report the detection of broad absorptions due to Si IV 4088-4116 A in the Luminous Blue Variable (LBV) AG Carinae during its last hot phase (2001-2003). Our NLTE spectral analysis, with the radiative transfer code CMFGEN, revealed the photospheric nature of these lines predicting, however, much narrower and deeper absorption profiles than observed. Using a recently-developed code to compute synthetic spectra in 2D geometry allowing for the effects of rotation, we could match the broad absorptions with a high projected rotational velocity of 190 +/- 30 km/s on 2001 April. Analysis of spectra obtained on 2002 March and 2003 January, when the star was cooling, yielded to a projected rotational velocity of 110 +/- 10 km/s and 85 +/- 10 km/s, respectively. The derived rotational velocities are proportional to R^-1, as expected from angular momentum conservation. We discuss the effects of such high rotation on the spectral analysis of AG Car, and on the wind terminal velocity. Our results show direct spectroscopic evidence, for the first time, that a LBV may rotate at a significant fraction of its break-up velocity. Thus, AG Car (and possibly other LBVs) is indeed close to the Gamma-Omega limit, as predicted by theoretical studies of LBVs.
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