The president also said that Chinese President Xi Jinping had ordered Chinese banks to cease conducting business with North Korean entities. Trump called the move "very bold" and "someone unexpected," and he praised Xi.
"I must tell you this is a complete denuclearization of North Korea that we seek," Trump said in brief public remarks during a meeting with the leaders of South Korea and Japan to discuss strategy to confront Pyongyang over its nuclear and ballistic missile programs.
Trump said the United States had been working on the North Korea problem for 25 years, but he asserted that previous administrations had "done nothing, which is why we are in the problem we are in today."
He added that the order will give Treasury Secretary Steve Mnuchin the "discretion to target any foreign bank knowingly facilitating specific transactions tied to trade with North Korea."
Trump's announcement came as he has sought to rally international support for confronting dictator Kim Jong Un's regime during four days of meetings here at the United Nations General Assembly. In a speech to the world body on Tuesday, Trump threatened to "totally destroy" the North if necessary and referred derisively to Kim as "rocket man."
In recent weeks, the U.N. Security Council has approved two rounds of economic sanctions but also left room for further penalties. For example, the sanctions put limits on the nation's oil imports but did not impose a full embargo, as the United States has suggested it supports. The Trump administration has signaled it also wants a full ban on the practice of sending North Korean workers abroad for payments that largely go to the government in Pyongyang.
Sitting down with South Korean President Moon Jae-in before the trilateral discussion with Japan, Trump said the nations are "making a lot of progress."
Moon praised Trump's speech to the U.N., saying through a translator that "North Korea has continued to make provocations and this is extremely deplorable and this has angered both me and our people, but the U.S. has responded firmly and in a very good way."
The Security Council had also applied tough new export penalties in August, and Secretary of State Rex Tillerson said Wednesday that there are signs those restrictions are having an economic effect.
“We have some indications that there are beginning to appear evidence of fuel shortages,” Tillerson said in a briefing for reporters. “And look, we knew that these sanctions were going to take some time to be felt because we knew the North Koreans...had basically stockpiled a lot of inventory early in the year when they saw the new administration coming in, in anticipation of things perhaps changing. So I think what we're seeing is a combined effect of these inventories are now being exhausted, and the supply coming in has been reduced.”
There is no sign, however, that economic penalties are having any effect on the behavior of the Kim regime and its calculation that nuclear tests and other provocations will ensure its protection or raise the price of any eventual settlement with the United States and other nations.
All U.N. sanctions have to be acceptable to China, North Korea's protector and chief economic partner. China's recent willingness to punish its fellow communist state signals strong disapproval of North Korea's international provocations, but China and fellow U.N. Security Council member Russia have also opposed some of the toughest economic measures that could be applied, such as banking restrictions that would affect Chinese and other financial institutions.
Anne Gearan in New York and Abby Phillip in Washington contributed to washingtonpost.com.