The nonlinear evolution of the internal m=1 kink instability in a pure Z-pinch (Bz identical to 0) is calculated by means of bifurcation theory, using an incompressible, ideal MHD model of the plasma. It is shown that, within the range of validity of this theory, all internal m=1 modes are stabilized nonlinearly. The analysis is carried out in detail for a pinch with a uniform current distribution, but the results in the regime of short axial wavelengths are probably relevant also for more general profiles. Thus, it is possible that nonlinear MHD effects may be of importance for the anomalous plasma stability observed in various fusion devices based on the Z-pinch.