The deal to evacuate Barzeh district mirrors similar agreements for opposition-held territory elsewhere in the country, allowing fighters safe passage in exchange for surrender.
It comes days after the start of a deal brokered by regime backers Russia and Iran and rebel supporter Turkey to put in place "de-escalation zones" where the government and opposition will halt hostilities.
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"Armed men and some of their families have begun leaving Barzeh on 40 buses heading towards northern Syria," state television said.
It added that the evacuation would continue for five days, but that rebel fighters who chose to stay could do so if they register with the government.
The channel did not specify how many people were expected to leave, but the Syrian Observatory for Human Rights monitor said up to 1,500 people - mostly rebels - would leave Barzeh today and head to the northwest province of Idlib.
The evacuation deal was struck late Sunday night, and dozens of people had gathered in Barzeh from the morning.
An AFP photographer saw rebel fighters carrying light weapons looking on as children and women in brightly-coloured headscarves pulled shabby suitcases and duffel bags.
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A source from the pro-government National Defence Forces said rebel fighters would be allowed to take their "personal weapons" with them.
Negotiations were ongoing for a similar deal in the district of Qabun, in Damascus's northeast, which forces loyal to President Bashar al Assad have been shelling heavily for weeks.
Speaking at a press conference in the capital, Foreign Minister Walid Muallem confirmed that the "reconciliation in Barzeh has begun."
"We are working on Qabun and there is the Yarmuk camp, where talks are underway for the evacuation of armed groups," he added.
Assad's government controls all but six districts of Damascus: Barzeh, Qabun, Jobar, Tishrin, Tadamun, and Yarmuk camp.
The evacuation is the first time that rebels and their families have left a district of the capital.
But the Barzeh deal follows a model of "reconciliation deals" that have been implemented across the country.
The government touts such deals as the best way to end the six-year conflict, which erupted in March 2011 with anti-regime protests and has killed over 320,000 people.
But the opposition says it is forced into them by heavy government bombardment and siege tactics.
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Monday's evacuation began as fighting subsided across the country after a deal signed by Russia, Iran and Turkey to implement four "de-escalation zones".
The agreement, inked in the Kazakh capital Astana, does not include Damascus city, and also excludes Sweida and Tartus, two provinces fully under government control.
It mandates an end to hostilities, including air strikes, in specified zones for six months.
The deal came into effect on midnight Friday, with fighting subsiding, but the co-sponsors have until June 4 to submit maps finalising the borders of the zones.
It calls for continued fighting against the Islamic State group, which is largely absent from the areas where the zones will be implemented, as well as former al Qaeda affiliate Fateh al Sham Front.
Fateh al Sham has a powerful presence particularly in Idlib province, which is included in the zones, where it is allied with other opposition groups.
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