With 97 percent of the state's precincts reporting, the Nebraska Secretary of State's office said Fischer -- who had been endorsed by 2008 Republican vice presidential nominee Sarah Palin -- had 41.08 percent of Republican votes cast, to 35.9 percent for Bruning.
The winner will face off against former U.S. Sen. Bob Kerrey in the general election in November.
Kerrey was headed for an easy victory in the Democratic primary Tuesday, with 79 percent of the votes cast.
The winner will succeed Sen. Ben Nelson, a Democrat who is retiring after his current term.
Thirty-five delegates to the Republican nomination convention in August in Tampa, Fla., were at stake in Nebraska. As of Thursday, CBSNews.com indicated Romney had 934 delegates to 92 for U.S. Rep. Ron Paul of Texas, who has suspended his campaign for the presidential nomination.
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