The Taliban accused the Afghan government and South Korean negotiators of failing to act in good faith after they rejected a list demanding eight named rebel prisoners be freed.
"Since Kabul's administration did not listen to our demand and did not free our prisoners, the Taliban shot dead a male Korean hostage," a Taliban spokesman said.
"If the administration of Kabul is not ready to release our hostages, then by 1 am (local time) the rest of the hostages will be killed," he said. "That time is the last deadline."
He said the Korean hostage had been killed in a desert area in the Qarabagh district of Ghazni close to where the 23 Korean church volunteers, 18 women and five men, were abducted on the main road south from Kabul last week.
Anxious family members of the Korean hostages gathered at the offices of a non-governmental agency in Seoul to follow developments on television. Sounds of crying emerged when news came out that one of the hostages had been killed.
Around 1,000 people gathered in suburban Seoul around Saemmul church, which sent the volunteers to Afghanistan, to pray for their safe return, broadcaster YTN reported.
James Bays, Al Jazeera's correspondent in Afghanistan, said that he was told by the spokesman that "a male hostage had been killed and his body was left next to the main Kabul-Kandahar highway."