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“Republicans are tired of having to hold their nose and go vote for Mike Castle,” said Ms. O’Donnell, who lost to Mr. Biden in 2008 and is still raising money for TV ads. “People don’t have to anymore.”

The primary is Sept. 14, and the winner likely will face County Executive Chris Coons, a Democrat, for the seat vacated by Mr. Biden.

Polls showing that Mr. Castle has less than 50 percent of the vote is another example that Republicans in Washington have failed to understand why voters have backed non-establishment, tea party candidates such as Sharron Angle in Nevada and Rand Paul in Kentucky, Ms. O'Donnell said.

She also has tried to inject urgency into her campaign by telling voters that the November winner will be immediately seated in Congress and that she, unlike Mr. Castle, would vote against cap-and-trade legislation.

“I’ll do more than vote,” she said. “I’ll advocate against it.”

A Castle staffer said the congressman has publicly stated that he is opposed to considering controversial legislation during a lame-duck session.