The dynamical consequences of a truncation in the disc of a spiral galaxy, like that suggested by the sharp decline of luminosity observed in the outer parts of some edge-on galaxies, are investigated, in relation to the interpretation of observed velocity curves. The disc truncation leaves a ‘signature’ on the rotation curve, in the form of a region of nearly constant velocity, followed by a steep decline of velocity just outside the truncation. Such a feature is clearly present in the observed rotation curve of NGC 5907, in which the luminosity truncation is also present. The observed velocity curve of NGC 5907 can be well reproduced by a two-component model, with a smooth spherical distribution of ‘dark mass’ (a halo) superimposed on the luminous disc. The best-fitting values for the mass in the disc and in the halo (inside the optical truncation) are 9 and 13.5 × 1010 solar masses, respectively. The mass-to-light ratio (luminosity in the J band) is about 11, in solar units. The model predicts the values of some quantities, such as the thickness of the gas layer, that could possibly be observed in the near future, thus providing a clear-cut test of the model itself.