LONDON — President Vladimir Putin's declaration of the annexation of four regions in eastern and southern Ukraine signals the onset of a new and highly dangerous phase in the seven-month old war, one that Western officials and analysts fear could escalate to the use of nuclear weapons for the first time in 77 years.

Putin has previously threatened to resort to nuclear weapons if Russia's goals in Ukraine continue to be thwarted. The annexation brings the use of a nuclear weapon a step closer by giving Putin a potential justification on the grounds that "the territorial integrity of our country is threatened," as he put it in his speech last week.

He renewed the threat on Friday with an ominous comment that the U.S. atomic bombing of Hiroshima and Nagasaki created a "precedent" for the use of nuclear weapons, echoing references he has made in the past to the U.S. invasion of Iraq as setting a precedent for Russia's invasion of Ukraine.

U.S. and Western officials say they still think it unlikely that Putin will carry out his threats. Most probably, they say, he is hoping to deter the West from providing ever more sophisticated military assistance to Ukraine while the mobilization of an additional 300,000 troops allows Russia to reverse or at least halt its military setbacks on the battlefield.

But the threats appear only to have strengthened Western resolve to continue sending weapons to Ukraine and the Ukrainian military is continuing to advance into Russian-occupied territory. Even as Putin was announcing the annexation in Moscow on Friday and newly conscripted Russian troops were arriving in Ukraine, Ukrainian troops were in the process of encircling Russian soldiers in the eastern city of Lyman, extending their reach from their recent advances in Kharkiv into the newly annexed region of Donetsk.

In all four regions that Putin said he was annexing - Donetsk, Luhansk, Kherson and Zaporizhzhia - Russia only controls part of the territory.

Now that the areas being fought over are regarded by Moscow as Russian, it is possible to chart a course of events toward the first use of a nuclear weapon since the 1945 atomic bombing of Japan.

"It's a low probability event, but it is the most serious case of nuclear brinkmanship since the 1980s" when the Cold War ended, said Franz-Stefan Gady, a senior fellow with the International Institute for Strategic Studies in London. "It is a very dangerous situation and it needs to be taken seriously by Western policymakers."

U.S. and European officials say they are taking the threats seriously. White House national security adviser Jake Sullivan said on Sunday that there would be "catastrophic consequences" if Russia resorts to the use of nuclear weapons. He refused to specify what those would be but said the precise consequences had been spelled out privately to Russian officials "at very high levels."

"They well understand what they would face if they went down that dark road," he said.

European officials say the threats have only strengthened their resolve to support Ukraine.

"No one knows what Putin will decide to do, no one," said a European Union official who spoke on the condition of anonymity to discuss a sensitive subject. "But he's totally in a corner, he's crazy . . . and for him there is no way out. The only way out for him is total victory or total defeat and we are working on the latter one. We need Ukraine to win and so we are working to prevent worst case scenarios by helping Ukraine win."

The goal, the official said, is to give Ukraine the military support it needs to continue to push Russia out of Ukrainian territory, while pressuring Russia politically to agree to a cease-fire and withdrawal, the official said.

And the pressure is working, "slowly," the official said, to spread awareness in Russia and internationally that the invasion was a mistake. India, which had seemed to side with Russia in the earliest days of the war, has expressed alarm at Putin's talk of nuclear war and China, ostensibly Russia's most important ally, has signaled that it is growing uneasy with Putin's continuing escalations.

But the annexation and the mobilization of hundreds of thousands of extra troops have also served as a reminder that the Western strategy hasn't yet worked enough to convince Putin that he can't win, said Alexander Gabuev, a senior fellow with the Carnegie Endowment for International Peace who was based in Moscow until earlier this year.

The West had been hoping that Ukrainian successes would force Putin to back down, but instead he is doubling down. "Time and again we are seeing that Vladimir Putin sees this as a big existential war and he's ready to up the stakes if he is losing on the battlefield," Gabuev said.

"At the same time I don't think the West will back down, so it's a very hard challenge now. We are two or three steps away" from Russia failing to achieve its goals and resorting to what was once unthinkable.

Those steps to secure its positions include Russia pushing hundreds of thousands more men onto the battlefield; escalating attacks on civilian targets and infrastructure in Ukraine; and perhaps also embarking on covert attacks on Western infrastructure.

Although the United States and its European allies have refrained from making direct accusations, few doubt that Russia was behind the sabotage of the Nord Stream pipelines in the Baltic Sea, said the E.U. official.

"I don't think anyone has doubts. It's the handwriting of the Kremlin," he said. "It's an indication of, 'look what is coming, look what we are able to do.' "

Nuclear weapons would only likely be used after mobilization, sabotage and other measures have failed to turn the tide, and it's unclear what Putin would achieve by using them, Gady said.

Despite some wild predictions on Russian news shows that the Kremlin would lash out at a Western capital, with London appearing to be a favored target, it is more likely that Moscow would seek to use one of its smaller, tactical nuclear weapons on the battlefield to try to gain advantage over Ukrainian forces, said Gady.

The smallest nuclear weapon in the Russian arsenal delivers an explosion of around 1 kiloton, one fifteenth of the size of the bomb dropped on Hiroshima, which would inflict massive destruction but on a more limited area.

Because the war is being fought along a vast, 1,500-mile front line, troops are too thinly spread out for there to be an obvious target whose obliteration would change the course of the war. To make a difference, Russia would have to use several nuclear weapons or alternatively strike a major population center such as Kyiv, either of which would represent a massive escalation, trigger almost certain Western retaliation and turn Russia into a pariah state even with its allies, Gady said.

"From a purely military perspective, nuclear weapons would not solve any of Vladimir Putin's military problems," he said. "To change the operational picture one single attack would not be enough and it would also not intimidate Ukraine into surrendering territory. It would cause the opposite, it would double down Western support and I do think there would be a U.S. response."

That's why many believe Putin won't carry out his threats. "Even though Putin is dangerous, he is not suicidal, and those around him aren't suicidal," said Ben Hodges, a former commander of U.S. Army Europe.

Pentagon officials have said they have seen no actions by Russia that would lead the United States to adjust its nuclear posture.