  Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract
perturbation of $T$ into an interval map with a hole. Given a number $\ell$,
$0<\ell<1$, we compute an upper-bound on the size of a hole needed for the
existence of an absolutely continuous conditionally invariant measure (accim)
with escape rate not greater than $-\ln(1-\ell)$. The two main ingredients of
our approach are Ulam's method and an abstract perturbation result of Keller
and Liverani.
