The statement, which was not posted on websites usually used by al-Qaeda, said Sarkozy was "thirsty for the blood of Muslim children, women and old people and eager to carry out the mission set by his masters in the Black [US White] House".
It said the campaign would target "all those who pursue the policies of the Black House".
The authenticity of the posting on the website could not verified.
The Abu Hafs al-Masri Brigades takes its name from an Al-Qaeda commander killed during the US-led invasion of Afghanistan in October 2001.
The group previously claimed responsibility for the July 2005 terror attacks in London, as well those in Madrid in March 2004 and in Istanbul in November 2003.
Just after the London bombings, the group warned European nations to pull their troops out of Iraq within a month or face more attacks like them.
The French government has not yet issued any public response to Tuesday's statement.