WASHINGTON (Reuters) - The Supreme Court upheld a key part of Arizona's crackdown on illegal immigrants on Monday, rejecting the Obama administration's stance that only the U.S. government should enforce immigration laws in the United States.

The nation's highest court, in an opinion by Justice Anthony Kennedy, unanimously upheld the state law's most controversial aspect, requiring police officers to check the immigration status of people they stop.

But in a split decision, the justices also ruled that the three other challenged provisions went too far in intruding on federal law, including one that makes it a crime for illegal immigrants to work and another that requires them to carry their documents.

"Arizona may have understandable frustrations with the problems caused by illegal immigration ... but the state may not pursue policies that undermine federal law," Justice Kennedy wrote in a 25-page opinion.

Kennedy said the mandatory nature of police checks did not interfere with the federal immigration scheme, and found unpersuasive the Obama administration's argument that this portion of the law must be preempted at this stage.

He said it was improper to block that provision before state courts had an opportunity to review it, and without some showing that its enforcement conflicted with federal immigration law. Kennedy also left open the possibility for future constitutional or other challenges to the law once it goes into effect.

The decision in part was an election-year setback for President Barack Obama, a Democrat. It went to the heart of a fierce national debate between Democrats and Republicans over the 11.5 million illegal immigrants the U.S. government estimates to be in the country.

Arizona Republican Governor Jan Brewer, who championed her state's measure, said, "Today's decision by the U.S. Supreme Court is a victory for the rule of law. It is also a victory for the 10th Amendment (of the U.S. Constitution) and all Americans who believe in the inherent right and responsibility of states to defend their citizens."

Obama has vowed to push for comprehensive immigration legislation if re-elected on November 6. Opinion polls show Hispanics, now equal to 16 percent of all Americans, overwhelmingly support Obama.

Republican presidential challenger Mitt Romney opposed the administration's challenge to the law.

Arizona, on the southwest border with Mexico, two years ago became the first of half a dozen U.S. states to adopt laws to drive illegal immigrants out. The high court's ruling cleared the way for other states to adopt similar laws.

About 360,000 of the country's illegal immigrants, or 3 percent, reside in Arizona. Most of the state's nearly 2 million Latinos are in the United States legally.

Obama this month announced an important change in federal immigration policy ahead of the election contest Romney, who has taken a tough stance against illegal immigration.

Hundreds of thousands of illegal immigrants who were brought into the United States as children could be able to avoid deportation and get work permits under the policy change announced by Obama. Most illegal immigrants in the United States are Hispanics.