South Carolina Democrat Bob Conley will face U.S. Sen. Lindsey Graham in November.
A recount Monday left the North Myrtle Beach engineer with a 1,058-vote lead over Mount Pleasant lawyer Michael Cone.
Cone told The Associated Press he is conceding to Conley. He says he may remain on the ballot as the nominee of the Working Families Party.
The recounts held Monday also left Florence Mayor Frank Willis one vote short of opponent Stephen Wukela.
In Spartanburg County, state Rep. Bob Walker lost the primary to challenger Joey Millwood by 19 votes. He is considering a protest. In Richland County, state Rep. John Scott beat Vince Ford by 75 votes for a state Senate seat.
The Election Commission meets Wednesday to certify the recounts.