The army and allied forces hold more than 75 per cent of east Aleppo, a rebel bastion since 2012, three weeks into their operation to capture all of the second city.
The Observatory said rebels had withdrawn from the last parts of the Old City under their control overnight after the army seized the neighbouring districts of Bab al-Hadid and Aqyul.
"Rebels were forced to withdraw from the Old City neighbourhoods of Aleppo for fear of being besieged," the Britain-based monitor said.
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Syrian army captures part of rebel-held Aleppo
The army has made steady gains since it began its latest bid to recapture east Aleppo.
It now controls all the areas east of the historic citadel, and parts of the Marjeh neighbourhood, the Observatory said.
Overnight, the army carried out heavy shelling of the Al-Zabdiya neighbourhood and other territory still under rebel control in the southeast of the city, the Observatory said.
It said at least 15 people, including a child, were killed in government fire on east Aleppo on Tuesday.
Three children were among 11 people killed by rebel fire on government-held areas of the city.
Fighting sees 80,000 flee east Aleppo
At least 80,000 people have fled east Aleppo since the Syrian army began an operation in mid-November to recapture the rebel-held sector of the second city, a monitor said Wednesday.
The Syrian Observatory for Human Rights said the figure included residents who have sought refuge in the government-held west of the city and a Kurdish-controlled enclave between the two sectors.
But it does not include others who have fled south to remaining rebel-held territory, Observatory chief Rami Abdel Rahman said.
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The army has made swift progress in its operation to recapture east Aleppo and now controls three-quarters of the sector, which had been a rebel bastion since 2012.
The government has been urging civilians to leave for months, and has accused the rebels of holding residents hostage for use as "human shields."
Before the assault began, an estimated 250,000 people remained in east Aleppo, which was surrounded by government forces in mid-July, halting deliveries of basic goods.
