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Tom Ross, chairman of Republican Party of Delaware, called Miss O'Donnell a liar who “couldn’t get elected dog catcher.”

Mr. Ross since has apologized and given his support.

However, Mr. Castle, a widely popular Delaware politician who has held public office in the heavily Democratic state since the mid-1960s, told debate sponsor CNN on Wednesday afternoon he would not endorse either candidate.

“Not because of the competence of any of the candidates, but because the primary I went through was very nasty in a variety of ways, both politically and personally, and I’ve just declined to get involved in that,” he said.