We present a detailed study of the formation of the amniotic sac in the avian embryo, and a comparison with the crocodile amniotic sac. We show that the amniotic sac forms at a circular line of stiffness contrast, separating rings of cell domains. Cells align at this boundary, and this in turn orients and concentrates the tension forces. The tissue fold which forms the amniotic sac is locked exactly along this line due to the colocalization of the stiffness contrast and of the tensile force. In addition, the tensile force plays a regenerative role when the amniotic sac is cut. The fold forming the ventral side of the embryo displays the same characteristics. This work shows that amniote embryogenesis consists of a cascade of buckling events taking place at the boundaries between regions of differing mechanical properties. Hence, amniote embryogenesis relies on a simple and robust biomechanical scheme used repeatedly, and selected ancestrally.