We study an intermittent map which has exactly two ergodic invariant
densities. The densities are supported on two subintervals with a common
boundary point. Due to certain perturbations, leakage of mass through subsets,
called holes, of the initially invariant subintervals occurs and forces the
subsystems to merge into one system that has exactly one invariant density. We
prove that the invariant density of the perturbed system converges in the
$L^1$-norm to a particular convex combination of the invariant densities of the
intermittent map. In particular, we show that the ratio of the weights in the
combination equals to the limit of the ratio of the measures of the holes.