Updated at 4:50 pm ET In a split decision, the Supreme Court on Monday upheld one part of a tough Arizona immigration law, but struck down other sections.

The part of the law the justices upheld requires police officers stopping someone to make efforts to verify the person’s immigration status with the federal government.

Text of the decision (PDF)

The justices struck down three other parts of the law:

• One making it a crime for an illegal immigrant to work or to seek work in Arizona;
• One which authorized state and local officers to arrest people without a warrant if the officers have probable cause to believe a person is an illegal immigrant;
• And one that made it a state requirement for immigrants to register with the federal government.

"Arizona may have understandable frustrations with the problems caused by illegal immigration" while the federal government tries to enforce immigration law, but the state "may not pursue policies that undermine federal law," wrote Justice Anthony Kennedy in the majority opinion.

The U.S Supreme Court annoucines decisions on June 25.

The court said that there are several ways for state officials to cooperate with the federal government on immigration enforcement, such as by responding to federal requests for information about when a particular non-citizen will be released from state custody.

"But the unilateral state action" to arrest and detain suspected illegal immigrants "goes far beyond these measures, defeating any need for real cooperation" between the state and the federal government.

Monday's decision is only a prelude to further litgation over what now remains of the Arionza statute.

Yuri Gripas / Reuters

Members of the media gather June 25 for a stakeout in front of U.S. Supreme Court in Washington.

NBC’s Pete Williams reported that “there are other lawsuits against this law. There are several civil liberties groups suing in Arizona, claiming that this law is racial profiling, and those cases have yet to work their way through the courts.”

Monday's decision was a partial victory for President Obama who had criticized the Arizona law, saying it “threatened to undermine basic notions of fairness that we cherish as Americans.” 

Republican presidential hopeful Mitt Romney said Monday, "I would have preferred to see the Supreme Court give more latitude to the states not less. And there are states now under this decision have less authority, less latitude to enforce immigration laws."

He added that immigration policy has "become a muddle. But it didn’t have to be this way. The president promised in his campaign that in his first year he would take on immigration ... put in place a long term program to care for those who want to come here legally, to deal with illegal immigration, to deal with securing our borders. All these things he was going to in his first year he had a Democrat house and a Democrat senate but he didn’t do it. And because he didn’t act, states and localities have tried to act and now the courts trying to get into it and sort things out and it’s a muddle"

Attorney General Eric Holder said in a statement that "I remain concerned about the impact" of the part of the law that requires police to attempt  to check the immigration status of people they stop when they have reason to suspect that the person is in the United States unlawfully.

Holder said the Justice Department "will continue to vigorously enforce federal prohibitions against racial and ethnic discrimination.  We will closely monitor the impact of S.B. 1070 to ensure compliance with federal immigration law and with applicable civil rights laws, including ensuring that law enforcement agencies and others do not implement the law in a manner that has the purpose or effect of discriminating against the Latino or any other community."

House Judiciary Committee Chairman Lamar Smith, R-Texas, said flatly that Monday's ruling was a defeat for those who seek stricter enforcement of immigration laws.

Smith said the decision "limits the ability of states to protect their citizens and communities from illegal immigrants. It is the federal government’s job to enforce our immigration laws, but President Obama has willfully neglected this responsibility. This dereliction of duty has left states to address the crime, job loss, and other costs of illegal immigration."

He added, “Unfortunately, under this Administration, today’s ruling essentially puts an end to immigration enforcement since the states no longer can step in and fill the void created by the Obama administration."

The Justice Department had moved quickly in 2010 to block enforcement of the law. The administration had argued that the Constitution vests exclusive authority over immigration matters with the federal government, not the states, and that where the federal government has pre-empted state action, no state can intrude on that federal turf.

The majority on Monday essentially agreed with that argument.

In the oral argument before the high court on April 25, Solicitor General Donald Verrilli said Arizona did not have the power to exclude or remove a person who is in the state illegally.

Although some critics of the law have contended that it would inevitably lead to targeting of Latinos simply because of appearance, speaking Spanish, or having a Spanish accent, Verrilli told the justices on April 25 “We're not making any allegation about racial or ethnic profiling in the case.”

Since enforcement of the law had been blocked by a federal judge soon after its enactment, the Obama administration did not present a record to the Supreme Court of the law leading to incidents of ethnic profiling of Latinos in the state.

Joining Kennedy's majority opinion were Chief Justice John Roberts and Justices Ruth Bader Ginsburg, Stephen Breyer and Sonia Sotomayor.

Justices Antonin Scalia, Clarence Thomas and Samuel Alito concurred in part and dissented in part.

Justice Elena Kagan, who served as Obama’s solicitor general, had recused herself from the Arizona case.

The high court’s decision comes just days after Obama announced a new administration policy of not deporting illegal immigrants under age 30 who came to the United States, or were brought to the United States before reaching age 16, who are in school, or have graduated from high school, gotten a general education certificate, or are military veterans. The illegal immigrants covered by the new administration policy will be permitted to apply for authorization to work legally in the United States. 

Ross D. Franklin / AP

Rep. Gabrielle Giffords and Ron Barber celebrate early Tuesday.

Updated at 12:54 a.m. ET: Democrat Ron Barber has won a special House election in southern Arizona to finish the term of former Rep. Gabrielle Giffords, The Associated Press projected.

Giffords, who was critically injured in a Jan. 2011 event in her district, resigned earlier this year.

A former aide to Giffords, Barber, who was also injured in the attack outside a supermarket that left six dead and 12 others injured, faced a challenge from Jesse Kelly, a Republican who narrowly lost to Giffords in the 2010 midterm elections.

The campaign was imbued with emotions that still linger from the attack, which forced Giffords’s retirement – despite remarkable progress toward recovery – earlier this year. But the special election campaign has also assumed a degree of political significance, given each party’s aggressive efforts to win the seat.

Bruising battle
The race evolved into a bruising battle between Kelly and Barber, fueled by hundreds of thousands of dollars of outside spending in the campaign. The district will be redrawn for this fall’s election, slightly in Democrats’ favor. But Giffords first won accolades for her political resiliency in a district that Republicans have won in the previous three presidential elections.

Democrats focused their resources on painting Kelly as an extremist who would seek radical changes to Medicare and Social Security even well beyond what most House Republicans had voted for in Wisconsin Rep. Paul Ryan’s budgets the past two years.

House Majority PAC, a Democratic super PAC, released one particularly effective ad featuring video of Kelly talking about eliminating corporate taxes and calling Social Security and Medicare Ponzi schemes.

Giffords herself made several public appearances in the past week – along with her husband, former astronaut Mark Kelly – pushing for her onetime aide.

A loss for Democrats would have threatened another demoralizing loss, though, on the heels of Republicans Gov. Scott Walker’s survival of a recall campaign last week in Wisconsin.

Both sides will inevitably spin the results of Tuesday’s primary as a harbinger for their chances come November, though the ultimate takeaways from this race might actually be more limited.

Testing ground for attacks
Democrats’ path toward retaking the majority in the House never ran through this seat, which they had controlled since Giffords first won in 2008. Barber will have to stand for re-election in November, and Republicans – including Kelly – vow to contest that race, too. But the district will favor Democrats more slightly, and it’s not clear how inclined that national Republicans will feel to invest heavily in that campaign.

But the race against Kelly provided Democrats in Washington a testing ground for their attacks against Republicans this fall associated with Medicare and Social Security.

Additionally, Republicans gained traction versus Barber by trying to tie him to House Minority Leader Nancy Pelosi and President Obama.

Barber had hedged for a period on whether he would support Pelosi as Democrats’ leader as elected, and he similarly punted on a May question about whether he would even vote for Obama. (Barber later clarified that he intended to vote for the president in November.)

But the fact that a Democratic candidate would feel the need to distance himself from Obama in Arizona raises questions about the viability of the president’s effort to win that state this fall. The Obama campaign has included Arizona in several of its Electoral College roadmaps; the state is rated Lean Republican in NBC News’ battleground map.

The Associated Press contributed to this report.