(Corrects location from south to north Lebanon in paragraph 2)

NAHR AL-BARED, Lebanon, Aug 13 (Reuters) - A sniper shot dead a Lebanese soldier on Monday during fighting between army units and al Qaeda-inspired militants at a Palestinian refugee camp, security sources said.

Lebanese troops are closing in on the remaining pockets held by Fatah al-Islam fighters inside the Nahr al-Bared camp in north Lebanon. At least 278 people have been killed in 12 weeks of fighting there, which has also displaced some 40,000 refugees and devastated most homes and other buildings.

The combatants traded fire in the narrow alleys of the camp. Troops fired artillery and tank shells at fortified bunkers used by the militants, who hit back with grenades and rockets. Four Katyusha rockets hit a civilian area outside the camp but no one was hurt, the security sources said.

The army had increased the pressure on the militants last week by dropping bombs onto their positions from helicopters.

Two soldiers were seriously wounded in the latest fighting. The death brought to 137 the number of soldiers killed in the battles. At least 100 militants and 41 civilians have also died.

In a clear sign that the militants were running out of options, senior Fatah al-Islam field commander Shahine Shahine made a conditional surrender offer, which the army rejected at the weekend, Palestinian sources said.

They said the army's command insisted any surrender should be unconditional, and that all members of the group must give themselves up. The fighting is Lebanon's worst internal violence since the 1975-1990 civil war.

Fatah al-Islam, which split from a Syrian-backed Palestinian faction last year, has Lebanese, Palestinians and other Arabs in its ranks, including some who have fought in Iraq. It says it supports al Qaeda's ideas, but has no direct links with it.

The conflict has further undermined stability in Lebanon, which has already been crippled by a prolonged political crisis and shaken by bombings that have killed six U.N. peacekeepers and two anti-Syrian lawmakers in the past eight months.

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