This sensor (1), for a fluid and suitable for use in a pipeline at least temporarily containing the fluid flowing therethrough, has at least one curved measuring tube (4, 5), which vibrates during operation and guides the fluid, and a housing enclosing the measuring tube. The housing is composed of a metal cap (7) of two cap halves (71, 72) and a supporting tube (6), in which the measuring tube is held at its inlet and outlet ends in a manner such that the tube can oscillate, and out of which a measuring tube segment protrudes sideways. Each cap half has an edge (73; 74) with, in each case, four edge portions (731, 732, 733, 734; 741, 742, 743, 744). The edge portions (731; 741) are straight, the edge portions (732, 733; 742, 743) are curved, and the edge portions (734; 744) have the shape of circular arcs. The edge portions (731, 732, 733; 741, 742, 743) are welded continuously to the supporting tube (6) and the edge portions (734; 744) are welded continuously to one another. The cap halves (71, 72) are cut out of a dish-shaped intermediate (70) having a formed, surrounding edge bead (701), provided with the shape of a quarter-torus by metal spinning.