Islamic militants in the Philippines have posted a video showing the beheading of a German hostage.
Juergen Kantner was abducted from his yacht off Malaysia's Sabah state in November. His companion Sabine Merz's body was later found on the boat.
A deadline for a 30m peso (£483,000; $600,000) ransom expired on Sunday.
Mr Kantner, 70, and Ms Merz had been abducted before. They were held for 52 days in 2008 by Somali pirates and were released after a ransom was paid.
The video, reported by the SITE militancy-monitoring group, shows Mr Kantner being killed by a knife-wielding man.
Government envoy Jesus Dureza confirmed the killing.
"Up to the last moment, many sectors, including the armed forces, exhausted all efforts to save his life. We all tried our best but to no avail," he said.
Abu Sayyaf is one of the smallest and most violent jihadist groups in the southern Philippines, known for its brutality, including beheadings.
It has pledged allegiance to so-called Islamic State and has carried out kidnappings of foreigners and Filipinos. Some have been released for ransoms but a number are still being held.
The German foreign ministry said in a statement that there was no longer any reasonable doubt that the German abducted in the Philippines was dead.
"We are deeply shocked by the inhuman and gruesome act," it added.
A Philippine police report said Mr Kantner had been killed in the Indanan area of southern Sulu province on Sunday afternoon.
However, no body has yet been recovered.
The Philippine military had carried out air strikes on Abu Sayyaf positions in Sulu over the weekend as the deadline approached.
After the German couple's yacht, the Rockall, was found off Laparan Island in Sulu province on 7 November, Abu Sayyaf had issued an audio message saying it was responsible for the abduction.
Ms Merz was apparently killed as she tried to resist the militants.
Analysis: Damien McGuinness, BBC News, Berlin
A year after Juergen Kantner was kidnapped off the coast of Somalia in 2008 he returned - to pick up his yacht and to carry on sailing in the region.
This decision, and the second abduction in November, sparked a debate in Germany about the extent to which individuals have a responsibility to avoid danger.
Commentators discussed whether it was brave or reckless to go sailing in some of the world's most dangerous seas that were plagued by pirates - particularly given that the German government had reportedly paid huge sums in ransom to release Mr Kantner and his partner the first time they were abducted.
But Mr Kantner justified his decision by saying that after 32 years on his boat he had no friends left in Germany, and that everything he owned was on his yacht.
"My boat is my life and I don't want to lose it," he said at the time. "I want to live and die sailing."