  I prove that general relativity admits no asymptotically well-behaved static
spherically symmetric traversable wormholes supported by classical scalar
fields and nonexotic matter. The theorem holds for all values of the scalar
field curvature coupling parameter $\xi$, even though fields with $\xi>0$ are
capable of violating the average null energy condition. Good asymptotic
behaviour (the effective Newton's constant being positive and finite at both
ends of the wormhole) can only be achieved by introducing additional exotic
matter, which must itself violate the null energy condition over some region of
nonzero volume. These results are insensitive to the number of scalar fields,
the form of their potentials, and the coupling between the fields and the
additional matter.
