  Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show
that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove
that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong
End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question,
whether $S$ can be infinite at all.
