The announcement came amid lingering uncertainty about the impact of Yemen's current political deadlock on the US counterterrorism campaign.
The slain cleric was identified as Sheikh Harith al-Nadhari, said to be in his 30s. Often seen in a white turban, he was among the group's few public faces frequently featured in online religious lectures. He also recently appeared in an al-Qaida video praising last month's deadly attack at the offices of French satirical weekly Charlie Hebdo.
Al-Nadhari was killed along with three others on Jan. 31, when a drone-fired missile blew up a vehicle in which the four were travelling in the southern province of Shabwa, according to the AQAP statement, posted on one of its official Twitter accounts.
There was no immediate comment from Washington.
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The Jan. 31 airstrike was the second U.S. Drone strike in five days, and was seen as signaling Washington's resolve to keep fighting the militants despite Yemen's political paralysis, brought on by a Shiite rebel power grab.
The prospect of a leaderless Yemen has raised concerns about Washington's ability to continue targeting Al-Qaida.
Led by Osama bin Laden's top aide Nasser al-Wahishi, al-Qaida's Yemen branch has posed the greatest danger to Western interests, especially the United States. After several unsuccessful operations on US soil, the group claimed the attack on the French magazine, saying it was meant to avenge cartoons depicting the Prophet Muhammad.