  We consider a point particle in one dimension initially confined to a finite
spatial region whose state is frequently monitored by projection operators onto
that region. In the limit of infinitely frequent monitoring, the state never
escapes from the region -- this is the Zeno effect. The aim of this paper is to
show how the Zeno effect disappears in the classical limit in this and similar
examples. We give a general argument showing that the Zeno effect is suppressed
in the presence of a decoherence mechanism which kills interference between
histories. We show how this works explicitly by coupling to a decohering
environment. Smoothed projectors are required to give the problem proper
definition and this implies the existence of a momentum cutoff. We show that
the escape rate from the region approaches the classically expected result, and
hence the Zeno effect is suppressed, as long as the environmentally-induced
fluctuations in momentum are sufficiently large and we establish the associated
timescale. We link our results to earlier work on the hbar -->0 limit of the
Zeno effect. We illustrate our results by plotting the probability flux lines
for the density matrix (which are equivalent to Bohm trajectories in the pure
state case). These illustrate both the Zeno and anti-Zeno effects very clearly,
and their suppression. Our results are closely related to our earlier paper
demonstrating the suppression of quantum-mechanical reflection by decoherence
