Context. Determining the gas density and temperature structures of protoplanetary disks is a fundamental task in order to constrain planet formation theories. This is a challenging procedure and most determinations are based on model-dependent assumptions. Aims. We attempt a direct determination of the radial and vertical temperature structure of the Flying Saucer disk, thanks to its favorable inclination of 90 degrees. Methods. We present a method based on the tomographic study of an edge-on disk. Using ALMA, we observe at 0.5″ resolution the Flying Saucer in CO J = 2–1 and CS J = 5–4. This edge-on disk appears in silhouette against the CO J = 2–1 emission from background molecular clouds in ρ Oph. The combination of velocity gradients due to the Keplerian rotation of the disk and intensity variations in the CO background as a function of velocity provide a direct measure of the gas temperature as a function of radius and height above the disk mid-plane. Results. The overall thermal structure is consistent with model predictions, with a cold (<12−15 K) CO-depleted mid-plane and a warmer disk atmosphere. However, we find evidence for CO gas along the mid-plane beyond a radius of about 200 au, coincident with a change of grain properties. Such behavior is expected in the case of efficient rise of UV penetration re-heating the disk and thus allowing CO thermal desorption or favoring direct CO photo-desorption. CO is also detected at up to 3–4 scale heights, while CS is confined to around 1 scale height above the mid-plane. The limits of the method due to finite spatial and spectral resolutions are also discussed. Conclusions. This method appears to be a very promising way to determine the gas structure of planet-forming disks, provided that the molecular data have an angular resolution which is high enough, on the order of 0.3−0.1″ at the distance of the nearest star-forming regions.