  We construct examples of static, spherically symmetric wormhole solutions in
general relativity with a minimally coupled scalar field $\phi$ whose kinetic
energy is negative in a restricted region of space near the throat (of
arbitrary size) and positive far from it. Thus in such configurations a "ghost"
is trapped in the strong-field region, which may in principle explain why no
ghosts are observed under usual conditions. Some properties of general wormhole
models with the $\phi$ field are revealed: it is shown that (i) trapped-ghost
wormholes are only possible with nonzero potentials $V(\phi)$; (ii) in twice
asymptotically flat wormholes, a nontrivial potential $V(\phi)$ has an
alternate sign, and (iii) a twice asymptotically flat wormhole which is
mirror-symmetric with respect to its throat has necessarily a zero
Schwarzschild mass at both asymptotics.
