Black smoke again billowed above the camp, now badly damaged by days of shelling, as the army tried to take out the group's firing positions.
"The Palestinian refugees are not being treated properly by Lebanon"
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A Palestinian source said efforts were under way to arrange a ceasefire that would put the army in control of the outskirts of the camp and leave fighters restricted to a small part of it.
Negotiations would then begin over the fate of the remaining Fatah al-Islam members.
A military source said: "We achieved the destruction of the Samed position on the northeastern side of the camp yesterday and the Lebanese flag now rises from there."
The Samed complex, which had been used as a weapons store and training centre, was one of Fatah al-Islam's main positions.
Three soldiers were killed and seven wounded in the latest battles, security sources said.
The army has slowly chipped away at the area controlled by Fatah al-Islam, without entering the camp.
Security forces are barred from going into Lebanon's 12 Palestinian refugee camps under a 1969 Arab agreement.
The fighting is Lebanon's worst internal violence since the 1975-1990 civil war. At least 153 people, including 71 soldiers, more than 50 fighters and 32 civilians, have been killed.
At least 27,000 of Nahr al-Bared's 40,000 residents have fled, mostly to the nearby Beddawi camp, since the fighting started on May 20.
The Lebanese authorities say the fighters must surrender, but Fatah al-Islam has vowed to fight to the death.
Fatah al-Islam emerged late last year after its leader, Shaker al-Abssi, and 200 fighters split from the pro-Syrian Palestinian faction Fatah al-Intifada (Uprising).
Lebanon's Western-backed government says Fatah al-Islam is linked to Syrian intelligence, a charge denied by Damascus and the group itself. Abssi has said he supports al Qaeda's ideas but has no organisational ties to Osama bin Laden's network.
Meanwhile, an explosion in the south Lebanon refugee camp of Ain al-Hilweh killed two people and wounded three others.
Shehadeh Jawhar, a leader of the Jund al-Sham group, was among the injured according to residents.
They said that the two dead men were the shop owner and his nephew. The owner is the uncle of Jawhar, who happened to be in the shop when the explosion occurred, they added.