A process for constructing a well (1) for exploiting an oil or gas reservoir, comprising the following operations: (A) drilling a formation submerged by a water head, at least 3,600 meters deep or more, reaching the formation from the surface of the water with a drilling riser (7), and a drilling tool which passes internally through the drilling riser; and evacuating through the drilling riser (7) at least one of the circulating drilling fluid, the oil or natural gas coming from the formations and the resulting drilling materials. The drilling riser (7) has an external diameter equal to or smaller than 17 inches and reaches a wellhead (3) having an internal diameter equal to or smaller than 18.75 inches, and positioned in correspondence with or close to the seabed submerged which covers the formation.