The Democratic Party of US President-elect Joe Biden is on the verge of taking control of the Senate as results come in from two elections in Georgia.
Raphael Warnock, a pastor from Atlanta, is projected to have won one of the seats. The other is on a knife-edge.
If the Democrats win both, Mr Biden will have a much better chance of pushing through his legislative agenda.
The election is being rerun because of Georgia's rule that a candidate must take 50% of the vote in order to win.
None of the candidates in November's general election met that threshold.
With 98% of votes counted, US TV networks and the Associated Press news agency called the first of the two races for Mr Warnock.
Control of the Senate in the first two years of Mr Biden's term will be determined by the outcome of the second run-off.
Mr Warnock is set to become the first black senator for the state of Georgia - a slavery state in the US Civil War - and only the 11th black senator in US history.
Claiming victory, he paid tribute to his mother, Verlene, who as a teenager worked as a farm labourer.
"The other day - because this is America - the 82-year-old hands that used to pick somebody else's cotton went to the polls and picked her youngest son to be a United States senator," he said.
Although the Democrats would need to take both seats to gain full control of Congress, the Republican party of outgoing President Donald Trump only needs to win one in order to retain the Senate.
When will we get a result?
Final results are expected by 12:00 on Wednesday, Georgia time (17:00 GMT).
The margins are extremely tight. Mr Warnock is projected to have won his race by 50.6% to 49.4% for Kelly Loeffler.
Edison Research, which supplies election results to news organisations including the BBC, projects Democrat Jon Ossoff has won the other seat by 50.2%, defeating Republican Senator David Perdue with 49.8%.
More than 98% of ballots from Georgia's 159 counties have now been counted. The remaining votes come from the Atlanta suburbs, which are projected to go heavily for the Democrats.
More than three million voters - about 40% of those registered in Georgia - voted before Tuesday. Early voting benefited Mr Biden in November's White House election.
Meanwhile, Mr Trump - whose unsubstantiated claims that he was the victim of electoral fraud left Republican strategists worried about turnout in Tuesday's Senate run-offs - continued to cast aspersions on the integrity of the vote in Georgia.
On Saturday, Mr Trump pushed Georgia's top election official Brad Raffensperger, a fellow Republican, to "find" enough votes to overturn Mr Biden's presidential election win in the state.
What's at stake in Georgia?
If both Democrats win, the Senate will be evenly split 50-50, allowing incoming Democratic Vice-President Kamala Harris the tie-breaking vote.
This would be crucial for pushing through Mr Biden's agenda, including on key issues such as healthcare and environmental regulations - policy areas strongly contested by Republicans.
The Senate also has the power to approve or reject Mr Biden's nominees for cabinet and judicial posts.
If Mr Ossoff and Mr Warnock both win, it would bring the White House, Senate and the House of Representatives under Democratic control for the first time since President Barack Obama took office in 2009.
What do the exit polls say?
Mr Trump's unproven claims of voter fraud may have eroded voter confidence in the election system, according to exit polls from Edison Research.
Its survey of voters leaving polling stations found around 70% were very or somewhat confident their votes would be counted accurately, a drop of nearly 15% from November's White House election.
Exit polls showed Georgians in a clean split over which party they want to control Congress: 49% favoured Republicans, while 48% said the Democratic party.
The demographics roughly matched those in November. Black voters, who made up 29% of the electorate, favoured the Democratic candidates nine-to-one. The Republicans, meanwhile, were winning a majority of white voters.
And these surveys showed that most voters were repeating the choices they made in November. Georgians who supported Mr Trump were casting ballots for Mr Perdue and Ms Loeffler, while Biden supporters were doing the same for Mr Warnock and Mr Ossoff.
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