Flow in a symmetric channel with a sudden expansion makes a transition from a symmetric flow to an asymmetric one due to a symmetry-breaking pitchfork bifurcation on a gradual increase of the Reynolds number if the system is perfectly symmetric. However, an unavoidable infinitesimal imperfection of the system may render the pitchfork bifurcation imperfect. A weakly nonlinear stability analysis is proposed to investigate the structural instability of the bifurcation for such a flow. As a result, an amplitude equation for a disturbance is derived by including the effect of the imperfection of the system, and its coefficients are evaluated numerically. The equilibrium amplitude of the disturbance is calculated from the amplitude equation and compared with the experimental results for the flow in a channel that is presumed symmetric and also with the numerical solution of the full nonlinear equations for the flow in a slightly asymmetric channel.