The claim of control of the former Yarmouk Palestinian refugee camp, 5 miles from the center of Damascus, came after severe fighting between rebels, supported by some Palestinians, and a Palestinian nationalist organization loyal to Syrian President Bashar Assad, the opposition said.
The rebel claims, which could not be independently verified, came after thousands of Yarmouk residents fled the former refugee camp, which is now a mixed Damascus neighborhood where more than 150,000 Palestinians live alongside 350,000 Syrians.
Hundreds of Palestinians fled Yarmouk Monday during an early morning lull in the fighting a day after the Assad regime killed at least 25 people in the neighborhood in airstrikes.
Their mass departure was part of an exodus that started after deadly fighting escalated Friday between increasingly hostile factions of pro- and anti-Assad Palestinians, The Wall Street Journal reported.
Palestinians historically have considered Assad a benefactor and an ally, but that has increasingly changed. For many Yarmouk Palestinians, Sunday's attacks ended the last vestiges of the regime's claim to be a Palestinian champion and protector, The New York Times said.
"I want to save my family's life," Yussef, a 40-year-old Palestinian refugee who hurried out of the camp with his family, told the Times. "I will never, ever return."
Some residents fled to neighboring areas while at least 5,000 fled to Lebanon, a Lebanese official told the Journal.
At least 22 busloads of people entered Lebanon from Syria in the past day, and a "majority were Palestinians fleeing Yarmouk," Lebanese Minister of Social Affairs Wael Abu Faour told the Times Monday.
The Yarmouk killings drew a rare condemnation from Palestinian Authority President Mahmoud Abbas, whose government earlier refrained from criticizing the Assad regime.
The regime continued shelling Yarmouk Monday, Rasim Abu Thawra, an activist who lives near Yarmouk, told al-Jazeera. The Times also reported regime shelling Monday.
The U.S. State Department expressed concern about the airstrikes. Spokeswoman Victoria Nuland called the bombardments "a significant and alarming escalation of the conflict in Syria."
The U.N. Relief and Works Agency for Palestine Refugees in the Near East said: "Credible reports point to civilian deaths, injuries and destruction of property in Yarmouk. There are also waves of significant displacement as Yarmouk residents, including UNRWA staff and their families, scramble to seek safety as the armed conflict persists."
The agency said it was housing more than 2,600 displaced residents at its facilities and in Damascus-area schools -- "and the number is growing rapidly," it said.
U.N. Secretary-General Ban Ki-moon called Sunday's airstrikes "a matter of grave concern."
Syrian Foreign Minister Walid al-Moallem told Ban in a phone call "terrorists" were behind Yarmouk's mayhem, state television and the official Syrian Arab News Agency reported. He also warned Palestinians not to offer "shelter or assistance to terrorist groups," SANA reported.
The Assad regime refers to insurgents as terrorists.
The SANA account said Syrian ground forces had refrained from entering Yarmouk but said nothing about the Syrian air and artillery attacks.
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