Aims.We aim to study the geometry and kinematics of the disk around the Be star α Arae as a function of wavelength, especially across the Brγ emission line. The main purpose of this paper is to understand the nature of the disk rotation around Be stars. Methods.We use the AMBER/VLTI instrument operating in the K-band, which provides a gain by a factor of 5 in spatial resolution compared to previous MIDI/VLTI 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.For the first time, we obtain direct evidence that the disk is in Keplerian rotation, answering a question that has existed since the discovery of the first Be star γ Cas by Father Secchi in 1866. We also present the global geometry of the disk, showing that it is compatible with a thin disk and polar enhanced winds modeled with the SIMECA code. We found that the disk around α Arae is compatible with a dense equatorial matter confined to 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 α Arae is rotating very close to its critical rotation. This scenario is also compatible with the previous MIDI measurements. Key words: