  We construct the supersymmetric effective action at order alpha'^4 of the
abelian open superstring. It includes the alpha'^4 terms in the abelian
Born-Infeld action, and in particular the leading derivative correction of the
form d^4F^4. Besides linear supersymmetry this sector of the open string
effective action also has a nonlinear supersymmetry. The terms d^4F^4 and their
fermionic partners have an arbitrary coefficient, and we discuss the possible
fate of such coefficients when higher orders in alpha' are included.
