In addition, Mr. Obama said the cycle of allowing North Korea to create a crisis and then receive incentives to halt its nuclear program must end.
"This is a pattern they've come to expect," the president said. "We are going to break that pattern."
Listen to the full press conference here:
The two presidents said a new U.N. resolution seeking to halt North Korea's development of nuclear weapons and ballistic missile must be fully enforced.
The U.N. Security Council unanimously approved a resolution Friday that imposed new sanctions against North Korea, expanded an arms embargo and allowed ship searches.
According to Lee, "under no circumstance are we going to allow North Korea to possess nuclear weapons."
President Obama said that North Korea's record of threatening other countries and spreading nuclear technology around the world means it should not be recognized as a legitimate nuclear power.
"We will pursue denuclearization on the Korean peninsula vigorously," he said. "So we have not come to a conclusion that North Korea will or should be a nuclear power. Given their past behavior, given the belligerent manner in which they are constantly threatening their neighbors, I don't think there's any question that that would be a destabilizing situation that would be a profound threat not only to United States' security but to world security."
North Korea has bargained with other countries for more than a decade about giving up its nuclear program, gaining such concessions as energy and economic aid, and then restarting its program. In late May, the North fired a series of short-range missiles and conducted its second nuclear test since 2006, prompting the latest U.N. resolution.
At the press conference, Lee also called on the North Korean government to release two jailed American journalists and one South Korean worker.
The two U.S. journalists were sentenced to 12 years of hard labor in North Korea for illegally crossing the border into the North and filming there, the official Korean Central News Agency said.
The two women -- reporter Laura Ling and editor Euna Lee -- were arrested in Kangan-ri in North Hamgyong Province, said the state-run agency. A third person, Current TV executive producer Mitch Koss, and their Korean-Chinese guide managed to flee, KCNA said.
