In an attempt to grab the high ground in the presidential ballot count, Al Gore issued a surprise offer last night to drop any legal challenge to the Florida election if hand recounts are permitted to go forward in three heavily Democratic counties.

But in a dizzying series of events, Florida's Republican secretary of state refused a few hours later to include the hand counts in her official tally.

And George W. Bush, one hour after that, summarily rejected Gore's proposal.

The election should be resolved not by "deals or efforts to mold public opinion," the Texas governor said, but "by the votes and by the law."

Gore, in the first public comment by either candidate after two days of deepening confusion and chaos in Florida, also offered to abide by the outcome of a statewide manual recount, if Bush preferred that alternative.

"The campaign is over, but a test of our democracy is now under way," the vice president said in a hastily announced statement just after 6:30 p.m. His running mate, Sen. Joseph I. Lieberman, was at his side.

Gore extended an invitation for a face-to-face meeting with his Republican rival before the results in Florida are final. That meeting would be intended "not to negotiate" but as a show of national unity, he said.

Bush rushed back to Austin from his central Texas ranch to deliver a response at 10:15 p.m. Like Gore, he read his statement over national TV from his official residence.

Both men projected a statesmanlike tone, saying the nation's interests should come before their own.

But Bush turned thumbs-down on Gore's proposal of a statewide manual count. He said it would make the vote tally "less accurate" because repeated handling of the ballots would introduce further error into the process.

Bush said the Gore proposal was nothing new, even though the vice president offered for the first time to abandon any legal action in connection with the election.

Bush also rejected the Democrat's offer to meet before the tally is complete. He did say he'd be glad to get together with Gore after the election is decided.

In line with efforts by his campaign to bring the dispute to an end by this weekend, Bush made it clear that he expected a winner in Florida to be known by Saturday, when Florida Secretary of State Katherine Harris intends to certify the results.

The vice president's unexpected offer was the first direct effort by either of the candidates to propose an end to the post-election season.

Gore addressed several of the concerns of politicians in both parties, including the fear that a bitter struggle over the election results could poison the atmosphere for whoever wins the presidency.

He urged his supporters and Bush's to "lift up this discourse, to refrain from using inflammatory language and to avoid statements that could make it harder for our country to come together once the counting is over." If he wins, Gore said, he'd be ready to travel to Texas to meet with Bush; if he loses, he added, he'd meet the governor wherever Bush wanted.

Gore also appealed to what polls show is the public's desire to see the matter settled through counting, rather than legal action. He said he would not take or support legal action to challenge the Florida count, if the manual recounts are allowed to continue.

And he spoke to those who want to see the matter resolved quickly by predicting that a "fair and final" outcome could be reached relatively soon under his proposal. Even a statewide hand recount, Gore said, should take no more than a week to complete.

"It's a smart tactic," said John J. Pitney of Claremont McKenna College in Pomona, Calif. "He is trying to seize the rhetorical high ground. On the other hand, he very much wants a process that will be very favorable to his election."

Gore trails Bush by 300 votes in the official vote tally in Florida, pending the count of overseas absentee ballots tomorrow.

Unless the manually recounted ballots are included, Gore is unlikely to win, many politicians on both sides believe. By most estimates, the traditionally Republican overseas ballots could add several hundred votes to Bush's current margin.

Democrats are hopeful that a hand recount of some 1.6 million ballots in the largely Democratic counties of Broward, Palm Beach and Miami-Dade would turn up enough additional votes for Gore to push him ahead of Bush. The Florida Supreme Court allowed the hand counts to go forward yesterday, but Harris, a co-chairwoman of the Bush campaign in the state, said last night that they would not be included in her final tally.

In addition, officials in Miami-Dade, after conducting a preliminary manual count, have decided not to recount the entire county. Democrats have challenged that decision, and Gore's offer may have been intended to put new pressure on county officials to change their minds.

National polls show that the public is willing to await a fair and complete count of the votes in Florida to determine the outcome of the presidential election.

Gore's offer "puts Bush in the box of defending what the American people think is indefensible, which is shutting off the count before the count is completed," said Paul Maslin, a Democratic pollster.

Both sides have indicated, by their words and actions over the past week, that they believe Gore might gain from a manual recount of the ballots, which many states, including Texas, require as a check on machine counts.

The Bush campaign has gone to court to try to block the hand counts. Unless the courts intervene, Harris could well certify Bush as the victor in Florida on Saturday, adding to a growing public perception that the Republican will be the next president, Maslin said.

Gore's offer to resolve the matter with a relatively swift count, with a promise not to drag out the matter in the courts, could be his best way of signaling to voters that he shares their desire for finality, the Democratic pollster added.

Gore said last night that his proposal would "settle this matter with finality and justice in a period of days, not weeks."

And, in his first public comments on the subject of manual vote counts, he maintained that "machines can sometimes misread or fail to detect the way ballots are cast." Gore added that "a careful hand count is accepted far and wide as the best way to know the true intentions of the voters."

Gore's offer to take the matter out of the legal arena is a sharp reversal of the position announced by his campaign last week, which was the first to threaten to back legal action.

Perhaps the main legal action that Gore would abandon, as a result of his proposal, is the challenges that have been filed to the "butterfly ballot" in Palm Beach County. Many legal analysts have suggested that those lawsuits are unlikely to result in any significant change in the count.

The Gore campaign sees the hand counts in Democratic counties as his best chance to take the lead in the Florida count. By offering to let his presidential ambitions rise or fall with the results of those hand counts in three counties, Gore is taking a calculated risk that he will gain enough votes to wipe out Bush's margin.

"I don't know what the final results will show," Gore said. "But I do know this is about much more than what happens to me or my opponent. It is about our democracy. My faith is in the people's will, in our Constitution and in our system of self-government."