"We announce our responsibility for the kidnapping of this evangelizing disbeliever who, through her work, has succeeded in driving large numbers of Muslim sons from Islam," Al-Akhbar quoted a spokesman for the group as saying in the video, which was not broadcast.
In return for her release, the jihadists called for the release of a number of their fighters jailed in Mali and one of their leaders, Abu Tourab, detained at the International Criminal Court at The Hague, according to the agency.
Abu Ahmad Tourab is the nom de guerre of Ahmad al-Faki al-Mahdi, who is accused of ordering the destruction of historical monuments in Mali's fabled city of Timbuktu in 2012.
Tourab, who was a leader of the Al-Qaeda-linked Malian group Ansar Dine, is the first jihadist to appear before the war crimes court.
An official from the news agency told AFP that Al-Akhbar had no plans to distribute the video.
It was the second time Stockly has been taken captive, following her kidnapping from Timbuktu in April 2012 by Islamist fighters.
The social worker, who has lived in Timbuktu for years, was said at the time to be the last Westerner living in the desert city, which she refused to leave when it fell to Islamist Ansar Dine rebels on April 1st that year.
Two weeks later, special forces from Burkina Faso swept into rebel-held northern Mali aboard a helicopter and whisked her to safety in a pre-arranged handover by Islamist rebels.
At the time a loose alliance of Tuareg and Islamist rebels took advantage of the political chaos in Mali's capital that followed a March coup by capturing the country's vast desert north, including Timbuktu.
Stockly's capture that year brought to 24 the number of hostages seized in the Sahel region, 20 of them held by Al-Qaeda's north Africa branch, Al-Qaeda in the Islamic Maghreb (AQIM) and another Islamist group, the Movement for Unity and Jihad in West Africa (MUJAO).
Almost all were subsequently released, but three foreign hostages seized, a South African, a Swede and a Romanian remain in captivity.
After Stockly's kidnapping in 2012, the Swiss foreign affairs department said it had discouraged her from another stay in Mali.
The jihadist fighters were chased from Mali's vast remote north in 2013 by a French-led military intervention.
A regional French counterterrorism force is still conducting operations in the area.
But entire swathes of the north remain beyond the reach of both the Malian army and foreign troops.
In November, 20 people, 14 of them foreigners, were killed in an attack claimed by jihadist groups on the Radisson Blu hotel in the capital, Bamako.
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