The invention concerns a method for casing a borehole which consists of successively lowering an unlimited plurality of strings of casing having a common diameter, the next one through the preceding one, to be installed one beneath the other and being sealingly connected. Thus a next casing (6) is lowered folded through the preceding casing (5) and the expansion process consists either in injecting pressurized fluid through the drillstring (54) towards the next casing (6) so as to re-inflate it, or the next casing (6) is a cylindrical pipe having a diameter smaller than the internal diameter of the preceding casing (5) and the expansion process consists in forcing a cylindrical gauge (52) from the top end (26) to the bottom bell (28) of the next casing (6) so as to increase its diameter to make it equal to the diameter of the preceding casing (5). The method can use a cementing shoe (34) and a metal-to-metal seal (29) between the casing string.