In the intervening time, the USA continues to put together for the following round of Chinese tariffs.
"We have fully signed and documented another very important part of the Immigration and Security deal with Mexico, one that the USA has been asking about getting for many years", Trump tweeted Monday, adding that it will "be revealed in the not too distant future".
Despite the confirmation that Mexico is willing to play its part in the deal, some critics pointed out that Trump may have been bluffing when he said there was an undisclosed segment under the border agreement. "It will be revealed in the not too distant future and will need a vote by Mexico's Legislative body!"
The president's warning to Mexico that there could be economic penalties for refusing to do something about the illegal immigration problem seemed to be more about political leverage than placing taxes on that country's imports.
Mr Ebrard said the U.S. side also wanted to designate Mexico as a "safe third country", which would have required Mexico to take in asylum seekers heading for the USA and process their claims on its own soil.
"It reflects diplomacy at its finest", Pompeo said during a news conference.
"It is a regional system", he added. "But if they're not, we're going to have to participate in this kind of discussion".
There was no sign of a new deployment in the southern border town of Tapachula, the entry point for the large migrant caravans that have crossed Mexico in recent months, said AFP correspondents.
A regional asylum compact like the one Ebrard described could have major implications for asylum seekers, said Sarah Pierce, an analyst at the nonpartisan Migration Policy Institute. Pompeo acknowledged that it was "something that we pressed for with them throughout the time of the negotiations". That demand was put on the table again by USA negotiators last week, but was not accepted by Mexico.
In a joint statement, they said both countries have taken "unprecedented" steps to curb irregular migration and human trafficking, to "strengthen bilateral co-operation" over border security, including "co-ordinated actions" and information sharing.
"France charges us a lot for the wine and yet we charge them little for French wine", he said, promising to "do something" about it.
Many migrants say they are afraid of returning to their home countries, which triggers a USA immigration court process that can take months or years to resolve due to backlogs in the system. Officials from those countries did not immediately respond to Ebrard's comments.
"For a long time, for many years, people tried to get what we got in a period of a couple of days, and they couldn't get it". Cuccinelli, a former attorney general of Virginia, has developed a reputation as a hardliner on immigration.
That marked the "most difficult" moment in bilateral ties since Mexican President Andres Manuel Lopez Obrador took office on December 1, 2018, Ebrard said.
Mexico rejected this demand, which would have forced the Mexican government to accept numerous migrants detained trying to enter the United States by making them submit their asylum claims in Mexico rather than with US authorities.