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HHS has asked for space to house up to 1,000 children at Lackland Air Force Base, and the government is trying to find even more facilities.

Meanwhile, Mr. Johnson has sent staff to southern Texas to make sure children are receiving medical care. He also has directed his department to develop “an aggressive public messaging campaign to outline the dangers of and deter” the children from trying to cross, the spokeswoman said.

“DHS is expanding awareness campaigns targeting potential crossers, in their home countries, in an effort to warn them of the extreme dangers associated with attempts to illegally enter the United States while also underscoring the fact that illegal crossers — including children seeking to reunite with families — are not eligible for legal status, including under prospective legislation,” Ms. Catron said.

But she acknowledged that the government has limited tools to stem the flow.

Jessica Vaughan, policy studies director at the Center for Immigration Studies, said the numbers have been rising steadily for several years, and that the administration should have been better prepared.

“This is a crisis on the level of the Mariel crisis. This far outstrips the agency’s capacity to deal with it in the normal way,” she said, referring to the 1980 mass emigration from Cuba. “But they saw this coming, too. They estimated months ago that it was going to be double the prior year and they don’t seem to be taking any steps to prevent it from happening.”

She said releasing the children into the community, where they live for years while awaiting a final decision on their cases, will encourage more families to send their children.

“I would argue that it would be perhaps more humane to deal with it firmly so that people stop taking the risk of putting children through this smuggling ordeal,” Ms. Vaughan said.

That is probably unthinkable for an administration that has carved most illegal immigrants in the interior of the U.S. out of danger of deportation, and is searching for more ways to halt deportations.

Left with few other options, the administration has pleaded for help from Mexico, which is the first to see the border crossers.

Secretary of State John F. Kerry raised the issue during a recent visit to Mexico City.

For now though, the administration continues to struggle.

Mr. Rosenblum of the Migration Policy Institute said he is able to come up with good policy answers to most immigration questions, even if they are not politically possible. But in this case, he said, that’s not true.

“On this one, there’s really not a good policy answer,” he said. “Anything you do to protect those kids creates perverse incentives for other families to send their kids, and anything you do on enforcement puts them back in those bad situations.”