  The one-loop low-energy effective action for non-Abelian N=4 supersymmetric
Yang-Mills theory is computed to order $F^6$ by use of heat kernel techniques
in N=1 superspace. At the component level, the $F^5$ terms are found to be
consistent with the form of the non-Abelian Born-Infeld action computed to this
order by superstring methods. The $F^6$ terms will be of importance for
comparison with superstring calculations.
