US President George Bush and the Afghan leader, Hamid Karzai, have vowed to put an end to the Taleban in Afghanistan.
Fears have been growing for the fate of the Korean hostages
After two days of talks in the United States, President Karzai said the Taleban were a defeated force which no longer endangered his government.
They said they would not bargain with the Taleban over 21 South Korean hostages being held in Afghanistan.
Two South Koreans from the abducted Christian group have been killed by the Taleban, who demand a prisoner swap.
A US presidential spokesman said there would be no "quid pro quo" over the captives - 18 of them women - who were seized on 19 July from a bus in Ghazni.
Following talks at Camp David, Maryland, White House spokesman Gordon Johndroe said: "Both leaders agreed that in negotiations for the release, there should be no quid pro quo for the hostages.
"The Taleban are brutal and should not be emboldened by this."
A man who claims to speak for the Taleban had earlier told the BBC that the hostages' fate was in the two leaders' hands.
He also said the Taleban would continue its kidnapping policy whether or not there was an exchange.
A demonstrator in Seoul protests against US policy in Afghanistan
In Seoul, a South Korean presidential spokesman said the government wanted to "work separately" from the Bush-Karzai summit to resolve the issue of the captive Christian aid workers.
About 100 protesters rallied near the US embassy in Seoul on Monday and handed in a letter addressed to Mr Bush.
At Camp David, Mr Bush and Mr Karzai presented a united front in their joint news conference and insisted progress was being made in Afghanistan.
President Bush said five million children - a third of them girls - were now going to school.
President Karzai said 85,000 children aged under five were alive thanks to healthcare improvements made since the Taleban regime was toppled in 2001.
But Mr Bush said he did not agree with remarks Mr Karzai made in an interview with CNN last Sunday, in which he said the Iranians were helping, rather than hindering, Afghanistan.
The US president said Tehran was "not a force for good" and vowed that the US would continue efforts to isolate it.
"I believe it is in the interests of all of us that we have an Iran that tries to stabilise not destabilise, an Iran that gives up its weapons ambitions and therefore we are working to that end," he said.
Bush and Karzai discussed civilian casualties
Praising his Afghan ally, Mr Bush said: "There is still work to be done, don't get me wrong. But progress is being made, Mr President, and we're proud of you."
Afghan security was the key issue in the leaders' two-day meeting, as well as the booming trade in illegal drugs, a resurgent Taleban and civilian killings.
Mr Karzai insisted the Taleban were not a long-term threat.
He said: "They're not posing any threat to the institutions of Afghanistan.
"It's a force that's defeated. It's a force that is frustrated. It's a force that is acting in cowardice by killing children going to school."
Mr Karzai said he had broached the subject of the growing number of civilians killed in US and Nato military operations in Afghanistan with the US leader.
He said: "He is as much concerned as I am, as the Afghan people are. I was very happy with that conversation."