Coriolis mass flowmeter with a measuring pipe (2) vibrating in coupled flexural and torsional modes, characterized in that an attachment (20) which is formed rotationally symmetrically with respect to an axis of rotational symmetry and can be made to undergo torsional oscillations of the same frequency, but opposite phase, in relation to the torsional oscillating modes of the measuring pipe (2) is mechanically connected to the measuring pipe (2) and the axis of rotational symmetry of the attachment runs parallel to the straight line (8) (central axis) defined by the center points (4a, 6a) of the cross-sectional areas (4, 6) of the inlet and outlet of the measured section or coincides with this line.