The truce, the latest in a series of local ceasefires in Damascus flashpoints, was struck on Monday.
The Syrian government forces and foreign-backed militants have been fighting for more than a year in and around several areas of the city.
In addition to Babbila, deals have been struck for local ceasefires in Qudsaya, Moadamiyet al-Sham, Barzeh, Beit Sahem, Yalda and Yarmuk Palestinian refugee camp.
They involve a ceasefire, a siege being lifted and food allowed to enter militant-held areas, with militants handing over heavy weapons and the government raising Syria’s national flag there.
Source: Al-Manar Website