  Aims. We aim to study the geometry and kinematics of the disk around the Be
star $\alpha$ Arae as a function of wavelength, especially across the
Br$\gamma$ emission line. The main purpose of this paper is to answer the
question about the nature of the disk rotation around Be stars. Methods. We use
the VLTI/AMBER instrument operating in the K band which provides a gain by a
factor 5 in spatial resolution compared to previous VLTI/MIDI observations.
Moreover, it is possible to combine the high angular resolution provided with
the (medium) spectral resolution of AMBER to study the kinematics of the inner
part of the disk and to infer its rotation law. Results. We obtain for the
first time the direct evidence that the disk is in keplerian rotation,
answering a question that occurs since the discovery of the first Be star
$\gamma$ Cas by father Secchi in 1866. We also present the global geometry of
the disk showing that it is compatible with a thin disk + polar enhanced winds
modeled with the SIMECA code. We found that the disk around $\alpha$ Arae is
compatible with a dense equatorial matter confined in the central region
whereas a polar wind is contributing along the rotational axis of the central
star. Between these two regions the density must be low enough to reproduce the
large visibility modulus (small extension) obtained for two of the four VLTI
baselines. Moreover, we obtain that $\alpha$ Arae is rotating very close to its
critical rotation. This scenario is also compatible with the previous MIDI
measurements.
