"There will not be a fifth term", Bouteflika said in a message carried by the official APS news agency, while suggesting that he would remain in office until his term expires on April 28.
Shortly after the announcements, crowds took to the streets in the capital, Algiers, waving flags as passing cars beeped their horns.
"France expresses its hope that a new dynamic that can answer the deep aspirations of the Algerian people will rapidly take hold", Foreign Minister Jean-Yves Le Drian said in a statement.
The Algerian government plane that brought President Abdelaziz Bouteflika to Geneva for medical treatment last month left Algerian airspace and headed north early on Sunday, flight radar applications showed.
"These kids don't have any political calculations", said a mother on the sidelines of the protest, Karim Ziad.
Army Chief of Staff Ahmed Gaid Salah added his voice, saying Sunday that the army and the people "have the same vision of the future".
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The news follows on the heels of bombshell reports , which said North Korea is rebuilding its main rocket launch site. He added that "One of the lessons... that both sides took away from the summit is that pressure works".
A number of protests have also been staged in Switzerland against the Algerian president since he arrived in the country.
Bouteflika, who suffered a stroke in 2013 and has rarely been seen in public, faces mass protests in Algeria over his plans to stand for a fifth term. Mr Bouteflika has ruled for 20 years. "They're just 13-year-old kids". Nearby, other young Algerians made V-for-victory signs. More than a quarter of Algerians under 30 are unemployed.
A declaration by judges yesterday that they would refuse to oversee the election seems to have been the final straw. Clerics said they would not accept government orders about what to preach.
The protests have surprised Algeria's opaque leadership and freed up Algerians, long fearful of the watchful security apparatus, to openly criticize the president.
He became president in 1999 and reconciled a nation riven by a deadly Islamic insurgency, but questions swirl over whether he is really running the country today. Algeria has stagnated for decades under veterans of the independence war who dominate the country.
The security forces have been mostly restrained during the demonstrations, a signal of the establishment's eroding willingness to keep the president in power through force.