Voters in the US state of Georgia have delivered their verdict in an election seen by many as a referendum on President Donald Trump.
Polling stations closed at 19:00 (2300 GMT), but results were not expected for several hours.
Early results give Republican Karen Handel a narrow lead over Democrat Jon Ossoff in the suburban Atlanta race.
Spending on all candidates has been put at $56m (£44m), making it the costliest congressional election in US history.
Democrats have suffered narrow defeats in Kansas and Montana this year.
Trump tweets support
Mr Ossoff narrowly failed to win the 50% needed to secure outright victory in the election for the Atlanta seat in April, forcing this run-off vote against Ms Handel.
The Democrats were looking to capitalise on the president’s low personal approval ratings to win Georgia’s sixth district seat.
On Monday, Mr Ossoff, 30, told local TV: “It’s a neck-and-neck race, and it’s all about turnout now. That’s why we’re so focused on getting out the vote.”
Ms Handel, 55, spent Monday campaigning across the district, and received support from a Donald Trump tweet on Tuesday.
The president said: “KAREN HANDEL FOR CONGRESS. She will fight for lower taxes, great healthcare strong security – a hard worker who will never give up! VOTE TODAY.”
In a second tweet, he said that Mr Ossoff was “weak on crime and security, doesn’t even live in district”.
Ms Handel came a distant second to Mr Ossoff in the April election, but the Republican vote had been split among 11 candidates.
Republicans believe the run-off will favour them in an affluent seat they have held since 1979.
They also believe last week’s shooting of Republicans on a Virginia baseball field will count in their favour at the polls.
The suspected gunman was a Democratic supporter and a Republican advert that sought to politicise the attack was condemned by Ms Handel.
The Atlanta seat was made vacant when Congressman Tom Price left to join the Trump administration as health secretary.
It is the type of seat the Democrats will need to win if they want to overturn the Republican House majority in mid-term elections next year.
There is another election on Tuesday, in South Carolina, but the Republicans are expected to hold the seat, which was vacated when Mick Mulvaney became Mr Trump’s budget director.
The Republicans have so far staved off defeats by the Democrats this year.
In April they narrowly defended a deeply conservative Kansas seat vacated when Mr Trump appointed Congressman Mike Pompeo to lead the CIA.
Last month, Republican Greg Gianforte won a special congressional election in Montana, despite being charged with assaulting a UK reporter.
Stakes are high – Anthony Zurcher, BBC News, Washington
If November was the height of political winter for Democrats, the party faithful have been desperately searching ever since for the first signs of spring. In special congressional elections in Kansas and Montana they’ve strained to see green shoots. Narrow defeats there, with talk of moral victories, have only gone so far.
In Georgia on Tuesday, if Democrats hope for a thaw, they need to post a win.
The Atlanta-area race is the kind of contest Democrats need if they want to take back control of the House of Representatives next year. It’s a historically Republican seat that Donald Trump barely won last November, full of the educated suburban voters who populate key swing districts in Florida, California and across the South.
Mr Trump has celebrated his party’s previous special election victories. Democrats would love to give him a taste of electoral defeat.
If they come up short, frustration – tinged with hopelessness – will follow. If they win, Republicans will look to next year’s balloting with foreboding bordering on panic.
It’s one election; a single seat out of 435. The stakes, however, are just that high.