Aims. The aim of this work is to study the structure of the protoplanetary disk surrounding the Herbig Ae star HD 163296. Methods. We used high-resolution and high-sensitivity ALMA observations of the CO(3–2) emission line and the continuum at 850 μm, as well as the three-dimensional Monte Carlo radiative transfer code, MCFOST, to model the data presented in this work. Results. The CO(3–2) emission unveils for the first time at submillimeter frequencies the vertical structure details of a gaseous disk in Keplerian rotation, showing the back and front sides of a flared disk. Continuum emission at 850 μm reveals a compact dust disk with a 240 AU outer radius and a surface brightness profile that shows a very steep decline at radius larger than 125 AU. The gaseous disk is more than two times larger than the dust disk, with a similar critical radius but with a shallower radial profile. Radiative transfer models of the continuum data confirm the need for a sharp outer edge to the dust disk. The models for the CO(3–2) channel map require the disk to be slightly more geometrically thick than previous models suggested, and that the temperature at which CO gas becomes depleted (i.e., frozen out) from the outer regions of the disk midplane is T < 20 K, in agreement with previous studies.