This paper presents new high angular resolution ALMA 1.3 mm dust continuum observations of the protoplanetary system AS 209 in the Ophiuchus star forming region. The dust continuum emission is characterized by a main central core and two prominent rings at r = 75 au and r = 130 au intervaled by two gaps at r = 62 au and r = 103 au. The two gaps have different widths and depths, with the inner one being narrower and shallower. We determined the surface density of the millimeter dust grains using the 3D radiative transfer disk code DALI. According to our fiducial model the inner gap is partially filled with millimeter grains while the outer gap is largely devoid of dust. The inferred surface density is compared to 3D hydrodynamical simulations (FARGO-3D) of planet-disk interaction. The outer dust gap is consistent with the presence of a giant planet (Mplanet ~ 0.7 MSaturn); the planet is responsible for the gap opening and for the pile-up of dust at the outer edge of the planet orbit. The simulations also show that the same planet could be the origin of the inner gap at r = 62 au. The relative position of the two dust gaps is close to the 2:1 resonance and we have investigated the possibility of a second planet inside the inner gap. The resulting surface density (including location, width and depth of the two dust gaps) are in agreement with the observations. The properties of the inner gap pose a strong constraint to the mass of the inner planet (Mplanet < 0.1 MJ). In both scenarios (single or pair of planets), the hydrodynamical simulations suggest a very low disk viscosity (α < 10−4). Given the young age of the system (0.5–1 Myr), this result implies that the formation of giant planets occurs on a timescale of ≲1 Myr.