A hollow precision-machined body, such as a circular waveguide for transmission of millimeter wavelengths, is welded to a metallic body with minimum distortion of the precision body, by directing a concentrated beam of heat-producing energy such as an electron beam or a laser beam or a plasma across and peripherally about the interface between the precision and metallic bodies to produce a bead whose width at the contact area is related to the clearance between the bead and the inner surface of the precision tube and the maximum allowable strain at the inner face of the precision tube. Alternately, the energy beam is directed along the interface and around the periphery thereof to produce a bead whose depth of penetration is related to the thickness of the precision body and the maximum permissible strain at the inner surface of the precision body.