WASHINGTON >> The White House today ordered all West Wing employees to wear masks at work unless they are sitting at their desks, an abrupt shift in policy after two aides working near the president — a military valet and Katie Miller, the vice president’s spokeswoman — tested positive for the coronavirus last week.

In an internal email obtained by The New York Times, people who work in the cramped quarters around the Oval Office were told that “as an additional layer of protection, we are requiring everyone who enters the West Wing to wear a mask or face covering.”

Asked at a Rose Garden news conference whether he had ordered the change, Trump — who did not wear a mask and has repeatedly said he sees no reason to — said, “Yeah, I did.” But officials said the new requirement was not expected to apply to Trump or to Vice President Mike Pence.

White House officials have scrambled since last week’s positive diagnoses to keep the virus from spreading throughout 1600 Pennsylvania Ave. even as the president, Pence and many other senior administration advisers who may have come into contact with Miller and the valet declined to self-quarantine. Trump said today that he and Pence had tested negative for the virus.

Three top public health officials have chosen to remain isolated for a period of time: Dr. Robert R. Redfield, the director of the Centers for Disease Control and Prevention; Dr. Stephen Hahn, the commissioner of the Food and Drug Administration; and Dr. Anthony S. Fauci, the director of the National Institute of Allergy and Infectious Diseases.

During Trump’s afternoon news conference, senior White House aides could be seen standing along the side of the Rose Garden — all of them wearing masks.

The White House also made some smaller changes, including displaying signs encouraging social distancing at entryways and asking aides during routine temperature checks if they are experiencing symptoms, according to officials. Monday’s email said that West Wing staff members were “not required to wear a facial covering while at their desk if they are appropriately socially distanced from their colleagues.”

Many of the president’s top aides and advisers have eschewed masks in their appearances with Trump, despite their own advice to their staffs to wear them. On Saturday, Defense Secretary Mark T. Esper and the top military chiefs sat barefaced around the table with Trump at the White House, apparently in contradiction of policy at the Pentagon, where officials have been social distancing for two months and wearing masks for several weeks.

Esper had made a point of extolling the virtues of wearing masks while indoors during a visit to U.S. Northern Command in Colorado earlier in the week, saying that “in the open air, it’s not as essential,” but adding that “in a room, we’re wearing masks.”

The new White House policy on masks came as Trump tried to reinvent his government’s troubled history on testing for the coronavirus, claiming that the United States was “unmatched and unrivaled” in its testing capacity but ignoring the early failures to provide testing that allowed the virus to spread invisibly for months.

Declaring once again that “if somebody wants to be tested right now, they’ll be able to be tested,” Trump said that his administration was working with states to allow them to conduct 12.9 million tests in May, insisting that the testing ability in the United States compares favorably with other countries.

“We are testing more people per capita than South Korea, the United Kingdom, France, Japan, Sweden, Finland and many other countries,” he said, ignoring countries where testing on a per-capita basis is higher, including Germany, Russia, Spain, Canada, Switzerland and at least 20 others, according to statistics compiled by Our World in Data.

Flanked by large posters that proclaimed “America leads the world in testing,” Trump also declared victory over the pandemic, saying that “we have met the moment and we have prevailed.” Later, under questioning, he revised his comments, saying he only meant to say the country had prevailed on increasing access to testing.

But the president’s claim that “we’ve prevailed on testing” was also premature, even by his government’s own standards. Though the United States has ramped up testing from 150,000 tests a day from a month ago to 300,000 a day recently, the current rate still remains far behind the 5 million daily target he set last month.

The president’s claim about testing being available for anyone was also misleading. It is one thing to have enough testing capacity for everyone who is symptomatic or has been in contact with someone who has tested positive, but that is quite different from having enough to provide reassurance to people considering returning to normal life.

The president announced that his administration had begun distributing $11 billion for testing approved by Congress almost six weeks ago and claimed that Germany and the United States were “leading the world in lives saved per 100,000.” That was also an exaggeration.

Germany does have one of the lowest mortality rates in the world, at 9.13 deaths per 100,000 people. The United States, by comparison, has a rate of 24.31 deaths per 100,000, according to data from Johns Hopkins University. That is lower than several other European countries, like Britain, Italy, Spain, France, Belgium and Sweden, but it is also higher than that of Canada, Japan, South Korea, Norway, Iran, Brazil and many others.

Trump also mischaracterized the trajectory of new cases in the United States.

“The numbers are way down from what they were two weeks ago,” the president said. “The numbers are really coming down very substantially. This weekend was one of the lowest we’ve had. The numbers are coming down very rapidly.”

The number of new cases reported across the country this weekend was 45,227. That is lower than the 60,872 cases reported from April 25-26. But over the past two weeks, the numbers of new cases overall have declined only slightly — not substantially.

Moreover, that trend is not “universal” across the country, as Trump claimed. While the figures in the New York City region have fallen, new cases are increasing in nine states and remained largely unchanged in more than two dozen.