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Dona Ana County Sheriff Todd Garrison told The Washington Times this week that the new monument will hurt his ability to patrol the region. He said it will create a corridor where smugglers and drug traffickers can operate with impunity.

In his statement Wednesday, Mr. Boehner said even Democrats realized there could be problems. The bill Mr. Udall wrote to designate the national monument included specific protections for border security.

“Yet the president and his administration have chosen to ignore those efforts and the concerns that its new restrictions will place additional burdens on Border Patrol personnel and limit access to high crime areas along the border, making it easier for drug smugglers and human traffickers to move in and out of the country,” Mr. Boehner said.

Democrats have accused Mr. Boehner of using Mr. Obama’s enforcement record as an excuse to forgo tackling immigration. They said there’s little more than a month left for Congress to act this year, before the intense campaign season takes over and no major legislation can pass.

“They keep finding excuses for inaction. There are no excuses. Enough is enough,” Sen. Robert Menendez, a New Jersey Democrat, said on the Senate floor Wednesday.

Democrats argue that if Mr. Boehner would allow a broad immigration bill to come to the House floor, it would pass.

Mr. Boehner disputes that and says too many Republicans don’t trust the president to enforce the laws on the books, much less a new immigration law that would require even tougher enforcement.