One Cambodian soldier was killed overnight, according to Cambodia, but a Thai military source near the border told AFP the incident was only a "misunderstanding" involving small arms fire.
He said 13 Thai soldiers were injured on Sunday night, one seriously, despite a ceasefire agreement on Saturday.
"The ceasefire only lasted 10 hours," says correspondent Arnaud Dubus. "On Sunday evening there was very heavy fighting for more than three hours. Shelling went almost 10 kilometres inside Thai territory and several schools and houses were destroyed."
On Sunday, Cambodia accused Thailand of damaging Preah Vihear, the 11th-century temple at the centre of the hostilities.
"Thailand does not invade any country. We protect our sovereignty," said Thai Deputy Prime Minister Suthep Thaugsuban. "The army has reaffirmed that retaliation was conducted carefully and avoided the temple.”
United Nations Secretary-General Ban Ki-moon on Sunday called on both sides to “exercise maximum restraint”.
Cambodian Prime Minister Hun Sen has urged the United Nations Security Council to hold a meeting and is warning that regional stability is at risk.
About 15,000 people spent the night at 38 temporary shelters, and 23 schools near the border have been told to close from Monday to Wednesday.
Cambodia said two of its soldiers and one civilian were killed on Friday, and Thailand said a villager on its side of the border also died. A Thai soldier was killed on Saturday. But local media in both countries say the death toll could be much higher.
The foreign minister of Indonesia, which is the current chair of the Association of Southeast Asian Nations, is expected in Cambodia on Monday, but Thailand says intervention by the regional bloc is "unnecessary".
The World Court ruled in 1962 that Preah Vihear itself belonged to Cambodia, although its main entrance lies in Thailand. The 4.6-square-kilometre area around the temple is claimed by both sides.
"There is political interest on both sides to use this fighting for nationalistic goals," says Dubus. "The Cambodian government is using the controversy to fuel nationalism."
Meanwhile, on the Thai side, the nationalist Yellow Shirt movement is using the dispute to put pressure on Prime Minister Abhisit Vejjajiva.
"The Thai government is cornered between the Yellow Shirts, who are trying to push it out because of this border controversy and the Red Shirts who are trying to push him out because of what happened last year," says Dubus.