An offshore oil-drilling rig comprising a deck mounted mobile on legs, each leg including a support base on the sea floor and a caisson open at its base and designed to penetrate into the sea floor. Each caisson is arranged inside the leg and is mobile along the axis of the leg between a retracted upper position and a projecting lower position at which it is anchored on the sea floor. The invention also concerns methods for installing such a rig on an oil-drilling site.