Abstract: A tubular blank (22) is rolled on a mandrel (14) in a pilgrim rolling mill. In a first rolling stage, a first part of the blank (22) is rolled on a first part (18,19) of the mandrel (14), with a reduction of the outside diameter of the blank (22) to the outside diameter of the guide tube and a reduction of the wall thickness of the blank (22) to one of a first wall thickness (e1) and a second wall thickness (e2) exceeding the first wall thickness (e1). The mandrel (14) is advanced in the axial direction. A second part of the blank (22) is rolled on a second part (20,21) of the mandrel spaced from the first part in the axial direction with a reduction of the outside diameter of the blank to the outside diameter of the guide tube and a reduction of the wall thickness of the blank (22) to the other of the first wall thickness (e1) and second wall thickness (e2).