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“The state’s whole complaint — the reason this law was passed and this case has arisen — is that the citizens of Arizona believe federal priorities are too lax,” he wrote. “The state has the sovereign power to protect its borders more rigorously if it wishes, absent any valid federal prohibition.”

Justice Scalia and Justice Alito said the majority opinion gives too much credence to the administration’s priorities rather than to the law itself.

The administration argued that it uses broad discretion to decide which illegal immigrants it wants to pursue, and said a state shouldn’t be allowed to interfere by setting its own penalties. But Justice Alito said that raises a situation in which Arizona’s law would be blocked under this administration, but could have been considered legal under a different administration.

“The United States’ pre-emption argument would give the executive unprecedented power to invalidate state laws that do not meet with its approval, even if the state laws are otherwise consistent with federal statutes and duly promulgated regulations. This argument, to say the least, is fundamentally at odds with our federal system,” he wrote.

Four sections of the Arizona law were under review: The court allowed police immigration status checks to proceed, but struck down the law’s provisions creating state penalties for illegal immigrants who try to get jobs, penalties for legal residents who don’t keep their papers with them, and new police powers to arrest those they think the federal government would be able to deport.

Effects unclear

Mrs. Brewer signed the legislation, known as S.B. 1070, two years ago, saying her state needed to act because the federal government wasn’t doing its job.

The Obama administration promptly sued and lower courts blocked four parts of the law. Mrs. Brewer appealed, and the Supreme Court held oral arguments in April.

As the case moved through the federal courts, a small number of other states moved ahead with their own crackdown laws. Officials in those states said they would have to study the court’s ruling but thought their own provisions granting police powers to check immigration status would pass muster.

Even as some states acted to crack down on illegal immigration, other states and localities passed laws and ordinances to codify so-called sanctuary-city policies that discourage police from reporting illegal immigrants to federal authorities.

Mr. Obama’s administration sued to stop the state crackdown laws, but has not taken action against the sanctuary-city policies.

During the debate over S.B. 1070, Mr. Obama and Attorney General Eric H. Holder Jr. criticized the legislation as opening up the chance for racial profiling. But when they sued, they didn’t make that argument, instead confining their challenge to issues of federal versus state power.

The racial profiling challenge could arise later, though, as the law begins to be enforced — something Mrs. Brewer, Arizona’s governor, said she expects.

“Our critics are already preparing new litigation tactics in response to their loss at the Supreme Court, and undoubtedly will allege inequities in the implementation of the law,” she said. “As I said two years ago on the day I signed SB 1070 into law, ‘We cannot give them that chance. We must use this new tool wisely, and fight for our safety with the honor Arizona deserves.’”

Last week, Mrs. Brewer issued an executive order asking that training materials be distributed to all police to refresh them on what constitutes reasonable suspicion for a stop. Race or ethnicity alone is not enough to meet the threshold.