Usually a Coriolis mass flowmeter consists of a fluid conveying vibrating pipe segment with a reflection symmetry, on which the time delay DELTAtau is measured between the vibrations of two symmetrically situated cross sections. For a homogeneous pipe segment, the proportionality factor K(c) between DELTAtau and the mass flowrate Q(m), i.e. the calibration factor of the instrument, is independent of the nature of the flowing fluid. Fixing a concentrated mass m(c) at the middle of the pipe segment - as required e.g. for the purpose of a symmetric excitation of the vibration - brings about a dependence of the factor K(c) on the fluid density. In the present paper the influence of the mass m(c) on the vibration spectrum and on flowmetering is investigated in detail for an instrument working with a straight pipe segment. It turns out that, whereas the frequency of the fundamental vibration mode is strongly influenced by m(c), the calibration factor K(c) is practically independent of the mass m(c), up to fairly high values compared to the mass of the fluid filled pipe segment.