Lebanese civilian refugees from the town were first ordered to flee the area by Israeli forces.
Moe's was founded by Moe Atallah, a Lebanese refugee who fled his homeland in 1976.
Over 100 Lebanese refugees were killed by the shelling of a UN base at Qana, in what the Israeli military said was a mistake.
Hotels and furnished flats all over Cyprus are overflowing with Lebanese refugees, many of them ironically refer to themselves as "the new boat people."
Apart from the emotional trauma of leaving one's country, getting out of Lebanon is difficult, expensive and above all dangerous, the Lebanese refugees say.
Hundreds of thousands more Lebanese refugees fled in panic today after Israel warned people in southern Lebanon and other areas to leave their homes.
The affected civilian population included both Lebanese and Palestinian refugees.
The meeting took place before either leader had learned of today's disaster, in which Israeli forces shelled a United Nations base crowded with Lebanese refugees, killing scores.
There too, of course, outstanding matters in dispute are the water problem and the situation of the Lebanese Palestinian refugees.
During the 1982 war in Lebanon, the IRC assisted Palestinian and Lebanese refugees.