The Supreme Court delivered a major victory on Monday to an anti-abortion group that sought to challenge an Ohio law that bans campaign statements deemed to be false.

The justices, in a unanimous decision, ruled that the Susan B. Anthony List can go ahead with a lawsuit challenging the law as a violation of free-speech rights.

Both liberal and conservative groups have criticized the law, saying it has a chilling effect on political speech. Even Ohio attorney general Mike DeWine declined to defend the law in court, sending his deputies to argue for the state.

The Susan B. Anthony List was accused of violating the law during the 2010 election, when it accused then-Ohio Democratic Rep. Steve Driehaus of supporting taxpayer-funded abortion because he backed the new health care law.

Driehaus threatened to take them before the Ohio commission that reviews the accuracy of political ads.

Writing for the court, Justice Clarence Thomas cited concerns about the chilling effect on groups wishing to run political ads.

"The credibility of that threat is bolstered by the fact that authority to file a complaint with the Commission is not limited to a prosecutor or an agency. Instead, the false statement statute allows 'any person' with knowledge of the purported violation to file a complaint," the opinion said.

Thomas said the existence of the law already has a chilling effect on political speech because people and interest groups have reason to believe their statements may be censured. The court warned that the law could impose "burdens" on "electoral speech."

The case began during the 2010 election when the Susan B. Anthony List planned to put up billboards ads attacking Driehaus. The ads accused Driehaus of supporting taxpayer-funded abortion because he supported President Obama's new health care law. Driehaus, a Democrat who opposes abortion, claimed the ads misrepresented the true facts and therefore violated the false speech law.

After Driehaus filed a formal complaint, the billboard owner feared legal action and declined to post the ads. The Ohio Elections Commission found probable cause that the ads violated the law, but Driehaus later dropped the case after losing his re-election bid.

When the Susan B. Anthony List challenged the state law as unconstitutional, a federal judge said the group didn't have the right to sue because the case was withdrawn and it hadn't suffered actual harm. The 6th U.S. Circuit Court of Appeals in Cincinnati agreed, and the Supreme Court reversed that.

Fox News' Shannon Bream and The Associated Press contributed to this report.