Late on Monday, Ennahar TV announced that the new prime minister was Noureddine Bedoui and said Ramtane Lamara was appointed to the new position of deputy prime minister. Some judges joined a lawyers' protest in the city of Bedjaia. "We must turn this crisis into a constructive process".
Bouteflika's announcement brought crowds back onto the streets in celebration on Monday evening.
"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.
Political sources said the military would nearly certainly play a leading role in the transition process, and was assessing three or four civilians who could be eligible to become leaders.
Brahimi, a well-known diplomat on the worldwide stage and an old friend of Bouteflika, could be viewed by the military as a guarantor of stability, according to political analysts. "We defeated the supporters of the fifth term!" said taxi-driver Mohamed Kaci, 50. The announcement that he meant to run for office for a fifth term triggered widespread protests in the country - now in their third week.
Bouteflika has managed to stay in power by pushing through constitutional amendments to keep extending term limits.
Jaguar that attacked selfie-taking woman won't be put down
One witness, Adam Wilkerson, told AZFamily .com that he and his mother heard the woman scream Saturday evening and ran to help. At some point shortly thereafter, the jaguar reached its two front paws outside the cage and grabbed the woman's arm.
Bouteflika vowed in his message "to hand over the duties and prerogatives of the president of the republic to the successor freely chosen by the Algerian people".
The judges' defiance comes as leading clerics refused to bow to government pressure to issue pro-government sermons, a further setback for the embattled president in the predominantly Muslim country.
Al Jazeera's Hashem Ahelbarra takes a look at Bouteflika's political life.
Algerian media and protest leaders expect Monday to be a decisive day for the gas-rich North African country after Bouteflika returned home Sunday from two weeks in a Swiss hospital.
The country's chief of staff said the military and the people had a united vision of the future.
The protests have surprised Algeria's opaque leadership and freed up Algerians, long fearful of the watchful security apparatus, to openly criticize the president. Lieutenant General Gaed Salah did not mention the unrest.