"He has a wealth of high-level experience with almost 450 professional games, making him an ideal addition to Bordeaux's attack," a club statement on their website read.
"His size, his goal-scoring talent and his fighting spirit will certainly help the Navy and White this season. We can't wait to see him on the pitch!"
Carroll, 35, had played in all four games for second-tier Amiens this season having joined them from English club Reading on a free transfer in 2023.
Bordeaux, once regulars in the Champions League, came 12th in Ligue 2 last season but financial turmoil meant they filed for bankruptcy in July and were demoted by France's National Directorate of Management Control (DNCG) to National 1.
Hopes of a takeover by Liverpool owners, Fenway Sports Group, were dashed and when Bordeaux surrendered their professional licence they dropped to the fourth-tier National 2.
Despite their fall from grace, Bordeaux attracted large crowds last season and the signing of Carroll, who once broke the transfer fee record for an English player when moving to Liverpool from Newcastle United for 35 million pounds ($46.32 million), will be seen as a positive step.
Carroll scored 54 Premier League goals during nearly 250 games combined for Newcastle, Liverpool and West Ham United and earned nine England caps but injuries often interrupted his career.
He could be in line for his debut on Saturday against Voltigeurs de Chateaubriant. Bordeaux have yet to win a league game this season, drawing two and losing one.
