A method of making steel undersea pipes designed to convey corrosive fluids including water, the method including acting in a workshop on land to assemble together respective ends of at least two unitary pipe elements with carbon-steel first welds to form pipe strings, the first welds being protected by first linings, and acting at sea from a laying ship to assemble together respective ends of the strings by use of anti-corrosion second welds of stainless steel or of anti-corrosion alloy to form a pipe. Each of the ends of the strings are provided with an anti-corrosion cladding which extends over the inside surface made of carbon steel of each end unitary pipe element over a first length of at least 4 times the thickness of the wall of the pipe.