(CNN) -- Arizona Gov. Jan Brewer vetoed a bill late Monday that would have required President Barack Obama and other presidential candidates to prove they were American citizens, born in the United States, before their names could have been placed on the state ballot.

The so-called "birther bill" got final approval in the state House last week. Now that Brewer, a Republican, has vetoed it, the bill will not become law unless legislators vote to override her veto.

"As a former Secretary of State, I do not support designating one person as a gatekeeper to the ballot for a candidate, which could lead to arbitrary or politically-motivated decisions," the governor wrote in a letter addressed to the Arizona House speaker.

Under the measure, if there were any dispute about whether a candidate had proved he or she had been born in the United States, Arizona's secretary of state would have the final say.

"This measure creates significant new problems while failing to do anything constructive for Arizona," she added.

Obama has been hounded by allegations since he began running for president in 2008 that he was not born in America. Critics contend, among other things, that he was born in his father's home country of Kenya. The U.S. Constitution stipulates that only "natural born" citizens are eligible to be president.

Obama has insisted that he was born in Hawaii, and the allegations against him have been repeatedly discredited in investigations by CNN and other organizations. Nevertheless, the issue remains politically potent among segments of the electorate and has served as a rallying cry for many of the president's opponents, most recently potential GOP presidential candidate Donald Trump.

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The author of the so-called "birther bill," Arizona State Rep. Carl Seel, has said the bill was not targeted at Obama, but at "maintaining the integrity of the Constitution."

Among other things, a candidate would have to show a copy of his or her birth certificate. If a birth certificate couldn't be produced, a candidate would have to show a combination of baptismal or circumcision records, hospital birth files, postpartum medical records or other documents. Candidates also would have to submit affidavits declaring their citizenship as well as sworn statements regarding their residency for the previous 14 years.

"I never imagined being presented with a bill that could require candidates for President of the greatest and most powerful nation on earth to submit their 'early baptismal or circumcision certificates' among other records to the Arizona Secretary of State. This is a bridge too far," wrote Brewer.

Fourteen other states are considering similar legislation this year, according to Jennie Bowser, a senior fellow with the National Conference of State Legislatures. Measures have failed in three states -- Connecticut, Maine and Montana.

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In an effort to counter the charges of the birthers, Obama's 2008 campaign produced a "certification of live birth," a document traditionally accepted legally as confirmation of a birth.

Both the current Hawaii governor, Neil Abercrombie, a Democrat, and the previous governor, Linda Lingle, a Republican, have insisted that Obama was born in their home state.

Nearly 75% of Americans believe Obama was definitely or probably born in the United States, according to a March 11-13 CNN/Opinion Research Corp. poll. More than four in 10 Republicans, however, believe the president probably or definitely was not born in America.