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(CNN) -- Democrat Chris Coons defeated Tea Party-backed Christine O'Donnell in Delaware's U.S. Senate race Tuesday, bursting Republican hopes of taking a seat that Vice President Joe Biden held for nearly four decades.

With an estimated 99 percent of precincts reporting, Coons led O'Donnell 56 percent to 40 percent.

The result will be a major disappointment for some establishment Republicans, who thought the candidate that O'Donnell defeated in the GOP primary -- former governor and current Rep. Mike Castle -- stood a better chance at taking the seat and edging the GOP closer to control of the Senate.

Coons, the executive of New Castle County, will succeed the retiring Democratic Sen. Ted Kaufman, who was appointed to replace Biden nearly two years ago.

"You sent a message that the politics of no, the politics of division, the politics of negativity have no place in this great state," Coons told supporters in his victory speech Tuesday night. "You said today that Delawareans want, that Delawareans deserve, not slogans but solutions to your concerns. ... And you want the people that you send to Washington ... to be optimistic about our nation and its future and further the belief that our best days are not behind us, but ahead of us."

O'Donnell, conceding the race Tuesday night, said she and her supporters still could celebrate, arguing they had shaken up Delaware politics for the better and had something to build on.

"We were victorious because the Delaware political system will never be the same -- that's a great thing. The Republican Party will never be the same, and that's a good thing," O'Donnell said. "Our voices were heard, and we're not going to be quiet now. This is just the beginning."

O'Donnell, backed by the Tea Party Express and endorsed by former Alaska Gov. Sarah Palin, surprised the GOP establishment by defeating Castle, a moderate, in the September primary. Polling had shown that Castle stood a strong chance of defeating Coons in a general election, and post-primary polling consistently showed Coons with a double-digit lead over O'Donnell, a onetime political consultant and conservative commentator who'd run for the office twice before.

O'Donnell lost Delaware's 2006 GOP primary for Senate before winning it in 2008 and losing to Biden in the general election. O'Donnell's primary victory this year thrust the race into the national spotlight, but her general election campaign got off to a rocky start, with GOP strategist Karl Rove questioning her credentials.

O'Donnell, 41, also had to contend with well-publicized statements from a decade earlier that she dabbled in witchcraft and questioned evolution theory. She made the comments during appearances on the program "Politically Incorrect" when she was a spokeswoman for conservative causes.

Her first campaign commercial began with O'Donnell declaring, "I am not a witch," in response to one of the statements, and she acknowledged that the resurfaced clips forced her to reinvent herself in the final weeks of the campaign.

Coons, 47, also was somewhat of a surprise candidate for Democrats, who had expected state Attorney General Beau Biden -- Joe Biden's son -- to run. When Biden decided not to seek the seat, Coons didn't face serious primary opposition.

Though polls showed her trailing Coons, O'Donnell kept hope until polls were closed, campaigning with an RV in the southern part of the state on Election Day before coming back to Dover, where she watched the returns.

At Dover High School, some voters hugged O'Donnell and others politely told her, "No," when she asked for their vote. There didn't seem to be too many voters who were making up their minds at the polling place.

CNN's Eric Marrapodi contributed to this report.