To conduct a fluid, the transducer has a flow tube which in operation vibrated by an excitation assembly. Inlet-side and outlet-side vibrations of the flow tube are sensed by means of a sensor arrangement. To produce shear forces in the fluid, the flow tube is at least intermittently excited into torsional vibrations about a longitudinal flow-tube axis. An internal portion of the transducer, formed at least by the flow tube, an antivibrator, the sensor arrangement, and the excitation assembly and mounted at least on the inlet and outlet tube sections, has a centroid which is located inside the flow tube. The transducer is suitable for use in viscometers or Coriolis mass flowmeter-viscometers. In spite of using only a single straight flow tube, it is dynamically well balanced in operation, and the development of bending moments by the torsionally vibrating flow tube is largely prevented. This also effectively prevents the transducer case or the connected pipe from being excited into sympathetic vibration. Measurement signals representative of mass flow rate are readily distinguishable from measurement signals representative of viscosity, particularly if the sensors used for the viscosity measurement are also used for the mass flow measurement.