A migrant caravan traveling from Central America through Mexico will not be permitted to enter the United States, the top American diplomat pledged in “a message” for “this caravan or any caravan which follows.”

“You will not be successful in getting into the United States illegally, no matter what,” Secretary of State Mike Pompeo said Tuesday. “I repeat: The caravan will not cross our border into our southern states under any circumstances.”

Pompeo has been working with the Mexican government and the United Nations agency responsible for refugees to arrange for the migrants to be stopped before they reach the Rio Grande. He challenged criticism that the Trump administration has taken a harsh line with refugees by lowering the number of annual admissions, but also defended a zero-tolerance policy for the current caravan.

“We are determined that illegal entry into the United States from this caravan will not be possible,” he said.

[Also read: Trump defends claim of Middle Easterners in migrant caravan, but concedes 'no proof']

To that end, Pompeo has been working with Mexican Foreign Secretary Luis Videgaray and the United Nations High Commission for Refugees, most recently in a phone call with Videgaray earlier Tuesday. “The United States stands ready to assist in these efforts,” the State Department said in a readout of the call.

Pompeo emphasized that the U.S. has been “historically generous” with respect to refugees and asylum-seekers, just as he did when announcing in September that the administration would lower the refugee cap from 45,000 to 30,000 in the coming fiscal year.

“The magnitude of this challenge is unequaled in any other country,” he said at the time. “We must continue to responsibly vet applicants to prevent the entry of those who might do harm to our country.”

He suggested Tuesday that this effort should not fall entirely on the U.S. “Those who want to come here legally have every means available to do that,” Pompeo said. “People can also file to be refugees in other countries other than the United States of America.”