  The leading terms in the tree-level effective action for the massless fields
of the bosonic open string are calculated by integrating out all massive fields
in Witten's cubic string field theory. In both the abelian and nonabelian
theories, field redefinitions make it possible to express the effective action
in terms of the conventional field strength. The resulting actions reproduce
the leading terms in the abelian and nonabelian Born-Infeld theories, and
include (covariant) derivative corrections.
