Last month, the council tightened sanctions against North Korea, unanimously adopting a resolution that Ms. Haley called “the most stringent set of sanctions on any country in a generation.”
But since then, North Korea carried out one of its most provocative missile tests in recent years, hurling a ballistic missile directly over Japan that prompted the government in Tokyo to warn residents in its path to take cover.
And on Sunday, the North conducted its most powerful nuclear test ever, with a blast that experts said was far more destructive than the bombs that the United States dropped on the Japanese cities of Hiroshima and Nagasaki during World War II.
In her remarks, Ms. Haley gave a lengthy summary of the North’s flouting of international law since 1993, when the United Nations urged the country to reconsider its decision to withdraw from the Nuclear Nonproliferation Treaty.
“Despite our efforts over the past 24 years, the North Korean nuclear program is more advanced and more dangerous than ever,” she said. “They now fire missiles over Japanese airspace.”
“They now have I.C.B.M. capabilities,” she said, referring to intercontinental ballistic missiles. “They now claim to have tested a hydrogen bomb. And just this morning there are reports that the regime is preparing for yet another I.C.B.M. launch.”
“We have taken an incremental approach,” Ms. Haley added, “and despite the best of intentions, it has not worked.”
Koro Bessho, the Japanese ambassador to the United Nations, also stopped short of threatening imminent military action, but said the danger from North Korea had been “raised to an unprecedented level” and was “a grave threat to the peace and the security of the world.”
Cho Tae-yul, the South Korean ambassador, described the North’s latest test as “do-or-die behavior.” He called for “truly biting and robust measures that Pyongyang finds really painful,” including blocking the flow of any money that might finance the North’s weapons program.
François Delattre, the French ambassador, also called for new sanctions. “It is no longer a regional threat, it is a global threat,” he said. “It is no longer a virtual threat, it is an imminent threat. It is no longer a serious threat, it is an existential threat.”
Liu Jieyi, the Chinese ambassador, vowed that Beijing would “never allow chaos and war” on the Korean Peninsula, where the United States and China were both combatants in a conflict that lasted from 1950 to 1953. He called for all sides to return to the negotiating table, as did Vasily A. Nebenzya, the Russian ambassador.
The warning by South Korean officials on Monday that the North may be preparing another ballistic missile launch did not make clear what sort of missile that might be.
The North first tested its new intercontinental ballistic missile, Hwasong-14, on July 4, and then did so again on July 28. The second test showed that the missile had a range of about 6,500 miles, which would put the western and central United States within range.
The nuclear test that the North carried out on Sunday set off a magnitude 6.3 tremor centered at the testing site in the country’s northeast, the United States Geological Survey said. It was followed by a weaker tremor believed to have resulted from a collapse in the testing site.
President Trump said on Twitter that Sunday’s test was an “embarrassment” to China, the North’s biggest ally and trade partner. He also criticized South Korea, an American ally, which he accused of “talk of appeasement.”
Then on Monday, the South Korean military carried out drills, with F-15K fighter jets and ground forces firing missiles in a simulated attack on the North’s nuclear site. The country’s president, Moon Jae-in, talked with Mr. Trump by telephone on Monday evening, the South Korean leader’s office said.
“President Trump reaffirmed the United States’ ironclad commitment to defend South Korea,” said Park Soo-hyun, a spokesman for Mr. Moon. “The two leaders also agreed to push for maximum pressure and sanctions against North Korea and a stronger sanctions resolution at the United Nations Security Council.”
During the call, the spokesman said, Mr. Trump resolved a major South Korean grievance by agreeing to let it build more powerful non-nuclear ballistic missiles.
Under a treaty with the United States that was aimed at preventing a regional arms race, South Korea has been able to build ballistic missiles only with a range of up to 497 miles. Those missiles’ payload were not allowed to exceed 500 kilograms, about half a ton.
On Monday, Mr. Trump agreed to lift the upper limit on the payload, Mr. Park said. Mr. Moon agreed to help the United States complete the deployment of a missile defense system, known as Thaad, as soon as possible.
Testifying before the National Assembly on Monday, Defense Minister Song Young-moo of South Korea said he had told his American counterpart, Jim Mattis, in a meeting last week that the United States needed to send long-range bombers, aircraft carriers and other strategic assets to the Korean Peninsula more often or regularly to reassure the South Koreans.
He said he had told Mr. Mattis that many in his country were calling for the reintroduction of American tactical nuclear weapons to South Korea. He did not disclose how Mr. Mattis responded.
Those calls came from South Korea’s main conservative opposition party and some domestic news media outlets after North Korea’s intercontinental ballistic missile tests in July. Recent surveys have shown that a majority of South Koreans either agree with reintroducing American nuclear weapons or support arming their country with nuclear weapons of its own.
But President Moon’s office said his government remained opposed to the introduction of American tactical nuclear weapons to South Korea, saying that doing so would make it more difficult to persuade North Korea to give up its own nuclear weapons.
On Monday, Ms. Haley told the Security Council that the United States would circulate a resolution for more sanctions against the North.
The most recent sanctions, imposed last month, restricted North Korea’s ability to export its most valuable natural resources, including seafood, coal and iron ore. The United States and its allies could tighten the screws further by seeking to block countries from sending oil to North Korea or accepting laborers from the North, an important source of revenue for the country.
Both would be likely to face pushback — Russia uses North Korean labor, for instance — though neither Russia nor China rejected the notion of additional sanctions in their comments to the Security Council on Monday.
Previous Security Council resolutions have taken weeks of closed-door negotiations among American and Chinese officials.