In this paper we investigate the uniqueness of solutions of the Ginzburg–Landau system for superconductivity in the regime where the thickness 2a of the film is small. We analyse the equations of first variation with respect to two shooting parameters and obtain estimates on all relevant quantities at the end of the film where the boundary conditions are prescribed. Using these estimates, we prove that the bifurcation curve of symmetric solutions is given by a decreasing function of the order parameter β when the film is thin enough. In addition, we prove that there is no curve of asymmetric solutions branching from the symmetric curve.