We develop exact solutions for the surface density and rotational velocity for a type of infinitely flattened, self-gravitating, truncated disc, whose rotational velocity approaches a constant value as the radius of truncation becomes large. This class of disc is both a generalization of Mestel's disc and a finite version of a member of Toomre's family of infinite discs, having index n = 0. Moreover, we show that all truncated, Toomre discs, having n > 0, can be generated by successive differentiations of the n = 0 results. For all finite Toomre discs, our exact solutions just beyond the truncation radii exhibit (usually narrow) regimes within which particle orbits would be epicyclically unstable. However, in each case, the instability can be removed by a small modification of the surface density in that region. We find that the nature of the truncation ‘signature’ for finite Toomre discs is qualitatively similar to those for truncated exponential discs (Casertano 1983). In addition, we examine the effects of adding spherical haloes to generalized Mestel discs, as well as to finite Toomre discs of index n = 1. Finally, we consider the nature of spiral density waves that might persist in generalized Mestel discs.