Abu Musab Abdulwadood, the named leader of the Salafist Group for Call and Combat, known by its French acronym GSPC, said that he was awaiting orders from Osama Bin Laden.
The French foreign ministry said that it was taking the threat seriously, according to the Algerian foreign ministry.
Güncelleme Tarihi: 20 Eylül 2018, 18:16
The Algerians said they too were monitoring the activities of the group.
GSPC recently announced links to al-Qaeda. France and America Abdulwadood said in the video posted online on Tuesday: "America and France are looting [Algeria's] treasures and taking control over our destinies after the thief Abdelaziz Bouteflika [the Algerian president] collaborated with them.
"Just learn, O Bouteflika, along with your aides, the generals and your Crusader masters, that we are coming with all God's might." It wasn't possible to verify the authenticity of the video message.
Excerpts of it were released on the internet on Saturday, and a full transcript of the 20-minute video appeared on Tuesday.
The group's leader, also known as Abdelmalek Dourkdel, vowed to press on with armed struggle and appealed to bin Laden for instructions.
Jean-Louis Bruguiere, the French anti-terrorism judge, told the I-tele news channel on Tuesday that such threats were not new: "For more than a year, now, we have known that the GSPC is allied with al-Qaeda and that one of the targets is France."
Jean-Baptiste Mattei, French foreign ministry spokesman, said the government "takes all terrorist threats very seriously".
GSPC operations have been confined to Algeria, but the group claimed responsibility for an attack near Algiers in November on employees of an affiliate of US company Halliburton.
French authorities have voiced concerns over potential GSPC cells in Europe.
Al-Qaeda announced its union with the GSPC for the first time in a video in September.
Ayman al-Zawahri, the group's number two, said he hoped the alliance would work against Western interests, singling out the US and France, which colonised Algeria.
The GSPC is said to number just a few hundred fighters, though scattered attacks blamed on the group are reported nearly every week.
It has sought to exploit international links after being cornered by security forces at home, analysts say.