We develop a nonlinear theory of a weak single bunch instability in electron and positron circular accelerators and damping rings. Assuming that the growth rate of the instability is much smaller than the frequency, we use a perturbation theory and averaging technique to derive a nonlinear equation for the amplitude of unstable oscillations with account of quantum diffusion effects due to the synchrotron radiation. Numerical solutions to this equation reveal a large variety of nonlinear regimes depending on the growth rate of the instability and the diffusion coefficient. Among the possible solutions we find regimes in which the instability amplitude saturates at a level that is proportional to the square root of the linear growth rate. In other regimes, the amplitude oscillate in a manner similar to the sawtooth oscillations. Comparison with the observation in the SLC Damping Ring at SLAC shows qualitative agreement with the patterns observed in the experiment. Show Partial Abstract