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Mr. Rauf called Cordoba House an opportunity for “healing, peace, collaboration, and interdependence” in his statement, but Mrs. Gellar said the effort to build what she called a “victory mosque” has caused enormous pain to families of Sept. 11 victims.

“It’s a grave insult, and the idea of this being outreach and healing and building bridges — frankly, it rings hollow,” she said. “It’s astounding, but it’s not surprising.”

One demonstrator during the New York vote held up a sign that said, “Islam builds mosques at the sites of their conquests and victories.” Another read, “Don’t glorify murders of 3,000. No 9/11 victory mosque.”

“I lost 3,000 American brothers and sisters, including courageous policemen and firemen, and this is a betrayal,” Linda Rivera, who held up the latter sign, told an Associated Press reporter through tears.

Critics noted that shortly after the terrorist attacks of Sept. 11, 2001, Mr. Rauf said that “United States policies were an accessory to the crime that happened” and cited his refusal to refer to Palestinian group Hamas as a terrorist organization.

Mr. Bloomberg celebrated the vote with a news conference with the Statue of Liberty as a backdrop.

“The World Trade Center site will forever hold a special place in our city, in our hearts,” he told reporters. “But we would be untrue to the best part of ourselves, and who we are as New Yorkers and Americans, if we said no to a mosque in Lower Manhattan.”

The New York Civil Liberties Union and the American Civil Liberties Union also praised the vote as “promoting our nation’s core values.”

“The free exercise of religion is one of America’s most fundamental freedoms,” they said in a statement Tuesday. “For hundreds of years, our pluralism and tolerance have sustained and strengthened our nation. … We see the center as a monument to pluralism, symbolic of America’s commitment to religious freedom.”

The American Center for Law and Justice, which opposed the mosque project, said after the vote that “we’re planning to file an Article 78 petition in state court to challenge the city’s actions. We will allege that there’s been an abuse of discretion in the Commission’s decision.”

The proposal had prompted months of contentious debate in New York and across the nation, with the highest-profile criticisms from former House Speaker Newt Gingrich and 2008 Republican vice-presidential nominee Sarah Palin. Both Republican primary candidates for New York governor — Carl Paladino and Rick Lazio — have said they would try to stop the mosque project, and former New York Mayor Rudolph W. Giuliani also has been critical.

Perhaps the most surprising criticism came last week from the usually liberal-leaning Anti-Defamation League. In a statement Friday, the Jewish civil rights group said that while the Muslim group had a legal right to build, the specific site is “counterproductive to the healing process.”