MIAMI (AP) – Rep. Katherine Harris, who as secretary of state oversaw Florida’s 2000 recount that gave George Bush the presidency, easily won the Republican nomination to challenge Senator Nelson, shrugging off critics who derided her campaign as spectacularly inept.

“Tonight I say to Bill Nelson: Come home, Bill. Enough is enough,” Ms. Harris said Tuesday.

Ms. Harris faces an uphill battle in her bid to unseat Mr. Nelson, though. Polls have shown the Democrat more than 30 points ahead of Ms. Harris in a general election matchup.

Ms. Harris became a darling of the Republican Party after the 2000 recount, and she parlayed her name recognition into two terms in Congress. But state GOP leaders tried to talk her out of running for Senate, citing fears she would lose to Mr. Nelson and spur a large turnout by Democrats in November that would hurt the entire Republican ticket.

In another closely watched race, Rep. Jim Davis won the Democratic nomination to succeed term-limited Gov. Bush, beating state Sen. Rod Smith. Mr. Davis’ opponent in November will be Florida Attorney General Charlie Crist, who claimed the Republican nomination.

Ms. Harris’ campaign was widely ridiculed, even by her own party. Fundraising lagged, her appearance was mocked, staff members kept quitting, and she was linked to a corrupt defense contractor.

Still, she won the primary comfortably, thanks to weak opposition and a strong base of support. Some 2 1/2 hours after the polls closed, the 49-year-old congresswoman arrived at her Tampa campaign headquarters to chants of “We want Katherine.”

“It’s a great victory because it shows each of us we can overcome adversity to achieve extraordinary victories,” Ms. Harris said.

With 99 percent of the precincts reporting, Harris had 49 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.

Mr. Nelson didn’t address Ms. Harris’ win directly but said in a statement that he looked forward to “spending the next six years continuing to fight for the people of Florida in the United States Senate.” The Democrat had no primary challenger.

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,” said state Division of Elections spokesman Sterling Ivey.

The Democratic race for governor tightened in recent days, but Mr. Smith fell short in his bid for a come-from-behind victory. Davis dogged Smith about his connections to big sugar companies, repeatedly pointing out how U.S. Sugar Corp. spent millions of dollars to fund attack ads.

Mr. Davis spoke to supporters in Tampa shortly after Mr. Smith phoned to concede.

“With all the talk of sugar in the news, let me say, how sweet it is,” Mr. Davis said. “It’s time to change direction, and tonight is a new beginning.”

Mr. Crist campaigned as a champion of consumer causes and the governor’s policies _ at least when it came to crime, taxes and education. A roar went up in Mr. Crist’s hotel suite in St. Petersburg when he told family and supporters he’d been declared the winner.

“All I want to be is the people’s governor,” Mr. Crist said, “and they should rest assured that if they elect me in November, no one will fight harder for the people.”

In other results, state Sen. Skip Campbell easily won the Democratic nomination for attorney general over a little-known lawyer who did not campaign. Bill McCollum, a former congressman, was unopposed for the Republican nomination.

Senate President Tom Lee won the Republican nomination for chief financial officer, setting up a November race against Democrat Alex Sink.

In the race for Ms. Harris’ House seat, auto dealer Vern Buchanan fended off four opponents for the GOP nomination to fill the 13th District seat. He will face banker Christine Jennings, who easily won the Democratic nomination.