The Korean war ended in a truce and not a peace treaty, which means that the US and Korea are still theoretically at war.
The conflict divided Korea into North and South.
While the tone remained playful on the surface, Bush made a firm response: "I can't make it any more clear, Mr President. We look forward to the day when we can end the Korean war. That will happen when Kim Jong-il verifiably gets rid of his weapons programmes and his weapons."
Under a February deal, North Korea agreed to give up its nuclear programmes. In return, Washington agreed to open talks on normalising relations with the North and look into removing a terrorism designation for Pyongyang.
North Korea shut down its main nuclear reactor in July, and US officials say Pyongyang has agreed to disable its nuclear programmess by the end of 2007. But Washington is suspicious the North may go back on its word.
Just hours after the awkward moment, Christopher Hill, US assistant secretary of state and chief US negotiator at the North Korean nuclear talks, announced a breakthrough in efforts to get Pyngyang to abandon its nuclear programme.
North Korea has invited nuclear experts from the US, China and Russia into the country to survey and recommend ways of disabling all of its atomic facilities by the end of the year, he said.
The team is to go next week.
Hill called the overture "another significant step toward the de-nuclearisation" of the Korean peninsula.Defence shield
In Friday's other developments, Bush talked to Vladimir Putin, the Russian president, about a planned US missile defence shield in Eastern Europe.
|"We look forward to the day when we can end the Korean war. That will happen when Kim Jong-il verifiably gets rid of his weapons programmes and his weapons"|
Putin, who opposes the plan, said that the two presidents agreed on a fresh meeting between the countries' technical experts to discuss a Russian-backed alternative for a joint system.
Bush did not comment.
Bush also had lunch with Southeast Asian leaders and invited them to his Texas ranch for a summit later this year.
In a speech to business leaders, the US president defended his policies in Iraq and urged Asia-Pacific nations to keep up the fight against "terrorism".
In the same speech, he said: "We must press the regime in Burma [Myanmar] to stop arresting, harassing, and assaulting pro-democracy activists for organising or participating in peaceful demonstrations."
The comments came a day after hundreds of Buddhist monks held a group of government officials for several hours and torched their cars in anger against the military.
Bush also said China should allow more freedoms in the run-up to the 2008 Olympic Games.
The summit, that formally starts on Saturday, is expected to focus on climate change and Bush used his speech on Friday to call for greater co-operation on the issue.
Bush acknowledged that some countries feared the US was trying to construct a successor to the UN's Kyoto Protocol outside of international efforts already under way.
"We agree these issues must be addressed in an integrated way," he said.
"We take climate change seriously in America."
The US has called for a September 27-28 conference in Washington of the 15 biggest polluters. A broader conference will be held at the UN in New York on September 24.