The nonlinear integral equation for the singlet distribution function, deduced from the first equation of the BBGKY hierarchy on the assumption that the difference in short range correlation between fluid and crystalline solid may be ignored, is solved near the bifurcation point for a system of hard spheres. The bifurcation point is the point at which crystalline solutions branch off continuously from the fluid solution, and has been recently obtained by Raveché and Stuart. The branch of the solution with face-centered cubic symmetry is shown to grow in a direction of decreasing density near bifurcation. This is not surprising, because freezing is a first order phase transition. It is argued that the crystalline state for this solution is unstable and the bifurcation point does not represent the metastability limit of the crystalline phase. Cases in which the integral equation has different kernels are also discussed in relation to the fluid instability.