Bifurcation characteristics of stably stratified plane Poiseuille flow have been investigated on a weakly nonlinear basis. It is found that the results are sensitive to the value of the Prandtl number, in that subcritical bifurcation persists for most values of the Prandtl number but is replaced by supercritical bifurcation over a range of small values of the Prandtl number. This range includes values characteristic of some liquid metals. The bifurcation becomes degenerate at a particular parameter set where the real part of the cubic nonlinear coefficient in the Stuart–Landau equation vanishes at criticality, and the situation is discussed by including higher-order terms in the manner of Eckhaus & Iooss (1989). An exact hyper-degenerate situation is also found to be possible for which the cubic and the quintic nonlinear coefficients lose their real parts simultaneously; this case is also analysed. For large values of the Prandtl number, stable stratification tends to promote subcritical instability.