With all precincts reporting, Ron Barber led Republican Jesse Kelly by nearly 7 percentage points, according to unofficial returns. Giffords, who survived an assassination attempt last year and resigned in January to concentrate on her recovery, had waited at Barber's election night party, anticipating victory.
"Well, a year ago I'd never have dreamt I'd be standing here, but life takes unexpected turns, and here we are," Barber told supporters late Tuesday.
"Gabby, we love you, and we are so grateful for your leadership and your dedication to this community," he said.
Kelly, 30, narrowly lost to Giffords in 2010. A few months later, she was shot in the head outside a Tucson grocery store while greeting constituents. Barber also was wounded, along with 11 others. Six people were killed, including a federal judge.
When Giffords relinquished her seat earlier this year, she asked Barber to run in the special election.
Barber will hold the seat until the end of the year. In November, both parties will vie to win a full two-year term.
From the start, national political bosses from both parties poured money into the politically moderate 8th District in an effort to influence the outcome. Republicans outspent Democrats more than 2 to 1, with much of the money paying for TV attack ads linking Barber to what the GOP called President Barack Obama's failed economic policies.
"Ron Barber is helping Obama, but he is hurting Arizona," one ad asserted.
Democratic leaders say they need to gain 25 seats to wrest control of the House from Republicans.
Although Giffords' endorsement was thought to carry persuasive emotional wallop, Democratic strategists took nothing for granted. In the hours before polls opened, party loyalists were urged to call 10 friends each to convince them to vote.
"We can't afford to lose Gabby's seat to a radical Republican who would go after everything that she stands for," read one release from the Barber camp.
Elsewhere, North Dakota voters resoundingly defeated an attempt to abolish the state's property taxes and were set to allow the University of North Dakota to rename its controversial mascot, which critics say denigrates American Indians.
More than 70 percent of voters rejected a grass-roots effort to eliminate state property taxes, according to unofficial returns, even though North Dakota has a budget surplus that exceeds $1 billion, in part due to an oil boom. More than 27,000 residents had signed a petition to get the measure on Tuesday's ballot.
"North Dakotans have a long history of rejecting extreme measures that take a big leap in policy," said Jon Godfread, a North Dakota Chamber of Commerce official who was part of a widespread coalition to defeat the measure.
Charlene Nelson, a stay-at-home mother of three who headed a citizen group called Empower the Taxpayer, said too many North Dakotans were losing their homes because of runaway property taxes.
"We started this movement before the oil boom," she said Tuesday. "But this isn't about being flush with oil money. It's based on principle. Property tax rates are rising faster than people's ability to pay them."
Nelson said the cuts could be made without raising other taxes. "Proceeds from the oil boom are $4 million a day - it's mind-boggling. We don't need this tax."
But opponents contended that eliminating property taxes would cost the state $812 million and leave no alternative but to raise taxes elsewhere.
North Dakota voters also empowered the University of North Dakota to drop its Fighting Sioux mascot, which the National Collegiate Athletic Association and others have called offensive.
But some American-Indian tribes had advocated keeping it, saying the name brought attention to the plight of local reservations.
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