Mexico’s president told President Obama Tuesday that his government will supply the documents necessary for millions of illegal Mexican immigrants to prove they’ve been living in the U.S. prior to 2010, a move that will help them qualify for Mr. Obama’s recently announced deportation amnesty.

In an Oval Office meeting, President Enrique Pena Nieto praised Mr. Obama’s executive action on amnesty as “very intelligent and audacious,” and said his government will do everything in its power to ensure that Mexicans who want to stay in the U.S. can do so.

The move has proven far more controversial in Washington, where Republican critics say the president’s unilateral move was unwise and unconstitutional.

Mr. Pena said his administration “is ready to give to the Mexican population living in the United States [support] so that they can show the documentation that is necessary to prove that they have been in the United States before 2010” and to have “all the other requirements” necessary to qualify for amnesty.

About two-thirds of the nearly 5 million illegal immigrants covered by Mr. Obama’s action are natives of Mexico. It’s not clear what kind of documentation the Mexican government will provide to show proof of residency in the U.S. prior to 2010 — the cutoff date to qualify for Mr. Obama’s amnesty plan and avoid deportation.

The Mexican president also said that Mexicans living in the U.S. will be able to obtain their birth certificates without having to travel to Mexico.

“They are going to be able to get this very important document,” he said.

In comments to reporters, Mr. Obama said U.S. officials “much appreciate Mexico’s commitment to work with us to send a very clear message around the executive actions that I’m taking,” actions the president said “are going to provide a mechanism so that families are not separated who have been here for a long time.”

Mr. Obama also praised Mexico’s help in stemming the surge of illegal child immigrants from Central America last summer on a smuggling route that used a key Mexican railroad.

“In part because of strong efforts by Mexico, including at its southern border, we’ve seen those numbers reduced back to much more manageable levels,” Mr. Obama said.

White House press secretary Josh Earnest said the U.S. has worked closely with the Mexican government “to enforce tighter border controls along their southern border and try to shut down the convenience of that transportation.”

“There is a pretty clear route that was taken by a lot of smugglers … across Mexico’s southern border,” he said. “It wouldn’t be a situation where we’re shutting anything down, but we certainly have raised our concerns about the way that … rail line has been used to transport a large number of … essentially people who are being trafficked along that rail line. And there are some steps the Mexicans have taken to reduce the instance of that tracking along that rail line.”

Mr. Obama also praised Mexico for cooperating in a campaign to warn residents in Mexico and other Central American nations that his amnesty action will not apply to anyone coming to the U.S. illegally in the future.

“We’re also going to be much more aggressive at the border in ensuring that people come through the system legally,” Mr. Obama said. “And the Mexican government has been very helpful in how we can process and message that effectively, both inside of the United States and in Mexico.”

The Mexican leader, now entering his third year in office, offered support for another recent Obama initiative, saying Mexico City would strongly support the diplomatic rapprochement between the United States and Cuba Mr. Obama announced last month. Mexico has long been a critic of the half-century-long U.S. embargo of Cuba.

“Mexico will be a tireless proponent of good relations between two neighbors,” Mr. Pena said, speaking in Spanish.

Following the meeting, the two governments released a list of strategic goals for 2015, pledging to collaborate in areas such as energy development, border security, trade and workforce development.

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