U.S. Rep. Katherine Harris overcame a campaign ridiculed even by her own party to easily claim the GOP nomination for the Senate on Tuesday, and Rep. Jim Davis held a narrow lead in the race for the Democratic nomination to succeed popular Gov. Jeb Bush.

Harris next faces an uphill battle against the Democratic incumbent, Sen. Bill Nelson, who had no primary challenger.

With 62 percent of the precincts reporting, Davis led state Sen. Rod Smith 46 percent to 42 percent. The winner will face Florida Attorney General Charlie Crist, who claimed the Republican nomination to replace Bush.

Crist had 64 percent of the vote to 34 percent for Tom Gallagher, the state's chief financial officer.

Harris had 50 percent of the vote against three relative unknowns. Attorney Will McBride ran second at 30 percent, and retired Navy admiral LeRoy Collins had 15 percent.

As Florida secretary of state six years ago, Harris oversaw the recount that gave George W. Bush the White House. She became a rising star in the Republican Party, parlaying name recognition into two terms in Congress.

But state GOP leaders tried to talk Harris out of running for the Senate, citing fears she would lose to Nelson while spurring a large November turnout by Democrats, which would hurt the entire Republican ticket.

Harris' campaign for the nomination was widely derided as spectacularly inept. Fundraising lagged, prompting her to pledge $10 million of her own money. Her makeup, clothes and personality were mocked on national TV. She was linked to a corrupt defense contractor. And staff members kept quitting in frustration.

Still, she won comfortably, thanks to weak opposition and a strong base of Republicans who loved her because of her role in the recount furor.

Despite a handful of late openings at polling places, the primary appeared to be debacle-free, with no problems reported to rival the troubled elections in 2000 and 2002. Rainy weather in South Florida and other parts of the state was expected to reduce turnout figures.

"The primary election in Florida today ran very smoothly," state Division of Elections spokesman Sterling Ivey said.

State Sen. Skip Campbell easily won the Democratic nomination for attorney general over a little-known lawyer who did not campaign. Bill McCollum was unopposed for the Republican nomination.

The Democratic race for governor tightened in recent days, with Smith trying to mount a come-from-behind victory.

Largely unknown when he entered the race, Smith touted his pro-agriculture positions and law enforcement background as a former state attorney. Davis dogged Smith about his connections to big sugar, repeatedly pointing out how U.S. Sugar Corp. spent millions of dollars to fund attack ads.

Crist campaigned as a champion of consumer causes and Bush's policies — at least when it came to crime, taxes and education. But Crist criticized the governor's decision to intervene in the Terri Schiavo right-to-die case, and said he wouldn't try to change the class-size limits that the governor has opposed.

Bush must step down because of term limits.