The decision was the third in as many terms to consider tough measures from Arizona addressing what lawmakers there say is a crisis caused by illegal immigration. But the Supreme Court has pushed back, protecting the dominant role of the national government in regulating immigration and voting.

Justice Antonin Scalia, writing for the majority in Arizona v. Inter Tribal Council of Arizona, No. 12-71, said a federal law requiring states to “accept and use” a federal form displaced an Arizona law requiring various kinds of proof of citizenship.

The federal law, the National Voter Registration Act of 1993, allows voters to register using a federal form that asks, “Are you a citizen of the United States?” Prospective voters must check a box for yes or no, and they must sign the form, swearing under the penalty of perjury that they are citizens.

The state law, by contrast, required prospective voters to prove that they were citizens by providing copies of or information concerning various documents, including birth certificates, passports, naturalization papers or driver’s licenses, that are available only to people who are in the state lawfully.

The state law was a result of a 2004 voter initiative, Proposition 200, that said it was meant to combat voter fraud. The law has given rise to tangled proceedings ever since. Under the Voting Rights Act of 1965, Arizona was required to obtain federal approval before it changed its voting procedures. The Justice Department granted approval in 2005.

According to the plaintiffs in the case, tens of thousands of Arizonans have been denied the ability to vote because they failed to present the required evidence.

Much of Justice Scalia’s majority opinion concerned the meaning of the phrase “accept and use.” Arizona officials argued that they do accept and use the form, but also require additional information. An airline may accept and use e-tickets, they said, but also require identification.

When the case was argued in March, Thomas C. Horne, Arizona’s attorney general, said the federal approach was insufficient to protect the integrity of federal elections in his state. “It’s essentially an honor system,” he said of the statement required by the federal form. “It does not do the job.”

In the decision on Monday, Justice Scalia said the phrase “accept and use,” when understood in context, meant that the federal form had to be accepted as sufficient. Chief Justice John G. Roberts Jr. and Justices Ruth Bader Ginsburg, Stephen G. Breyer, Sonia Sotomayor and Elena Kagan joined all of the majority opinion, and Justice Anthony M. Kennedy joined most of it.

In a long dissent, Justice Clarence Thomas said the Constitution gave states the power “to determine the qualifications of voters in federal elections, which necessarily includes the related power to determine whether those qualifications are satisfied.”

“Congressional legislation of voter qualifications was not part of the framers’ design,” Justice Thomas wrote.

In a second dissent, Justice Samuel A. Alito Jr. focused on the language of the federal law, which he said was ambiguous. The majority’s interpretation of it, Justice Alito wrote, “produces truly strange results.” He said he would read the law to mean that states “accept and use” the federal form so long as it is “a meaningful part of the registration process.”

Justice Alito likened his proposed process to the common application used by many colleges and universities. Those institutions, he said, “also require that applicants submit various additional forms and documents.”

Justice Scalia wrote that Arizona had additional options if it wished to obtain documentary proof of citizenship. It may ask the Election Assistance Commission, a federal body, to make changes to the federal form.

Arizona made such a request in 2005, and the commission split 2 to 2, effectively rejecting it. The state did not challenge that action in federal court. The commission recently approved a request from Louisiana to require additional information from its voters, Justice Scalia noted. He said Arizona could ask again.

In dissent, Justice Alito said the majority was giving the state an empty promise. He pointed out that the commission “currently has no members, and there is no reason to believe that it will be restored to life in the near future.” In response, Justice Scalia suggested that the state could sue in federal court based on its inability to obtain relief from the commission.

Last year, a divided 10-judge panel of the United States Court of Appeals for the Ninth Circuit ruled that the federal and state laws “do not operate harmoniously” and “are seriously out of tune with each other in several ways.” The court blocked the state law.

The decision from that panel effectively affirmed a 2010 ruling from a three-judge panelthat included Justice Sandra Day O’Connor, who retired from the Supreme Court in 2006 but occasionally acts as a visiting appeals court judge. She joined the majority in ruling that the state law was inconsistent with the federal one and so could not survive.

Justice O’Connor was in the Supreme Court’s courtroom on Monday to see the announcement of the decision