This article deals with an auction mechanism for airspace slots, as a means of solving the European airspace congestion problem. A disequilibrium, between Air Traffic Control (ATC) services supply and ATC services demand are at the origin of almost one fourth of delays in the air transport industry in Europe. In order to tackle this congestion problem, we suggest modifying both pricing and allocation of ATC services, by setting up an auction mechanism. Objects of the auction will be the right for airlines to cross a part of the airspace, and then to benefit from ATC services over a period corresponding to the necessary time for the crossing. Allocation and payment rules have to be defined according to the objectives of this auction. The auctioneer is the public authority in charge of ATC services, whose aim is to obtain an efficient allocation. Therefore, the social value will be maximized. Another objective is to internalize congestion costs. To that end, we apply the principle of Clarke-Groves mechanism auction: each winner has to pay the externalities imposed on other bidders. The complex context of ATC leads to a specific design for this auction.