An elated President Bush claimed a reelection victory yesterday after a tumultuous night of vote counting and a gracious concession by challenger John F. Kerry, and he pledged that he would seek to earn the trust of those who did not back him during the long, contentious campaign.

In an explicit appeal to those Americans who voted for Kerry, Bush said: "To make this nation stronger and better, I will need your support, and I will work to earn it. I will do all I can do to deserve your trust. A new term is a new opportunity to reach out to the whole nation."

Bush spoke to jubilant supporters at the Ronald Reagan Building in Washington, where he had planned to go for a pre-dawn victory speech after he had won Florida's 27 electoral votes and appeared to have locked up Ohio's 20 votes. He postponed that event when Kerry declined to concede the election overnight and signaled a possible fight over the vote totals in Ohio.

But an hour before Bush's appearance, an emotional Kerry took the stage at Boston's historic Faneuil Hall to offer Bush his congratulations and a formal concession. The Massachusetts senator had called Bush earlier to convey the same message privately. Kerry snuffed out the hopes of many Democrats who were eager to keep the fight for the White House alive by declaring, "We cannot win this election."

Bush will begin his second term with strengthened majorities in the House and Senate. With GOP candidates picking off a string of Democratic open seats, Republicans expanded their Senate caucus from 51 to 55 members -- a significant gain but still not a filibuster-proof margin. Senate Minority Leader Thomas A. Daschle (S.D.) lost his reelection bid to former congressman John Thune (R). In the House, the GOP added three seats and could emerge with a 29-seat majority once all the races are concluded.

With the second term that eluded his father secured, Bush pivoted to the task of trying to heal a nation that appeared on Tuesday as culturally and geographically divided as the country that produced the disputed presidential election in 2000. Vice President Cheney said that Bush had run on a clear agenda and that "the nation resounded by giving him a mandate."

Bush's speech offered an olive branch to the opposition, but he provided no hint of policy concessions to the Democrats. He outlined a domestic agenda that included broad tax reform and a proposal to allow younger workers to establish personal accounts with some of their Social Security payroll taxes. Many Democrats oppose his Social Security plans, and he may face partisan opposition on tax reform.

The president also vowed to continue to put the fight against terrorism at the forefront of his agenda, saying, "With good allies at our side, we will fight this war on terror with every resource of our national power so our children can live in freedom and in peace."

His stance on terrorism proved to be a significant political asset on Tuesday, but Bush faces enormous problems in trying to stabilize Iraq and pull off elections there scheduled for early next year. In his speech, the president did not mention the frayed international relationships that also will occupy him now that the election is over.

Bush claimed 51 percent of the popular vote to Kerry's 48 percent, with a margin of about 3.5 million votes, removing the label of minority president that he had carried since 2000. Four years ago, Bush lost the popular vote to Vice President Al Gore, but on Tuesday he became the first president since his father in 1988 to be elected with a majority of all votes cast. Independent Ralph Nader proved to be a non-factor, winning less than 1 percent.

With Ohio in his column, Bush won 30 states and 279 electoral votes. Kerry won 19 states and the District for 252 electoral votes. Iowa and its seven electoral votes remain in doubt. Bush was leading there with 100 percent of precincts reporting, and while counties were still tabulating absentee and provisional ballots, officials in the state said they did not expect a change in the lead.

Two states -- New Hampshire, which went for Kerry, and New Mexico, which went for Bush -- switched sides from 2000, despite efforts by both sides to take the campaigns into each other's territory.

Nearly 120 million Americans voted, or about 60 percent of those eligible, the highest number since 1968, according to the Associated Press. Many strategists believed an increase of that magnitude would favor Kerry, but the Bush campaign proved more than equal to the task of getting supporters to vote.

The swift and courteous end to the campaign came in marked contrast to the emotional roller coaster that played out overnight and that provided eerie similarities to the triggering events that produced the 36-day recount in Florida four years ago.

The battle for Ohio turned out to be short and conclusive. By the time more than 90 percent of the precincts there had reported, Bush strategists were certain there was no way for Kerry to win the state, and they chafed that the challenger would not concede.

Kerry aides originally believed there might be enough provisional ballots -- those cast by voters whose eligibility was in doubt -- to win Ohio. At that point, Kerry's running mate, Sen. John Edwards (N.C.), made a speech at Boston's Copley Plaza in which he vowed that "every vote would be counted," a thinly veiled warning that the Democrats were prepared to begin legal action to contest the state. At the time, Kerry aides said, there was pandemonium inside the campaign.

Overnight, the Kerry campaign's senior staff, in a series of calls with the boiler-room leadership in Washington and political and legal advisers in Ohio, analyzed the situation. They concluded that the estimated 150,000 provisional ballots were not enough to overcome Bush's margin of 136,000 votes in Ohio, even if Kerry were to win the lion's share of them.

Some lawyers argued that Kerry had a good legal argument to make and said that if the campaign was serious about a possible challenge, it needed to move immediately to force the state's counties to adopt uniform rules for counting the provisional ballots. Eventually, senior adviser Tad Devine said, the Kerry high command presented the candidate with a unanimous recommendation not to fight the count. "It's fair to say the unanimous recommendation was that this would not succeed," he said.

Kerry further discussed the situation with Edwards, Sen. Edward M. Kennedy (D-Mass.) and campaign manager Mary Beth Cahill, eventually agreeing that it was time to concede. At 11 a.m. yesterday, Kerry called Bush in the Oval Office to concede the election and pledge to bridge the nation's divisions. Three hours later, accompanied by his wife, Teresa Heinz Kerry, he left his home in Boston's Beacon Hill area for the short drive to Faneuil Hall.

There, he found a hall packed with campaign staff members and supporters, many of them trying to hold back tears over a loss that they never dreamed possible as they heard results of the first wave of exit polls Tuesday afternoon.

Kerry wasted no time in ending any talk of contesting the election. "In America, it is vital that every vote count and that every vote be counted," he said, in a nod to the exhortation that Edwards had invoked almost 12 hours earlier and that the two had used to rouse the Democratic base throughout the campaign. "But the outcome should be decided by voters, not a protracted legal process."

Kerry choked back tears and his voice broke as he recalled the experiences of his two-year campaign and talked about the need for unity in the election's aftermath, citing his conversation with the president. "We talked about the danger of division in our country and the need, the desperate need, for unity and for finding the common ground, coming together," he said.

Kerry advisers fully expected to win the election, based on their final polls, their analysis of Bush's weaknesses, their belief that the country hungered for change and their confidence that they would do a better job than the Republicans of getting their supporters to vote. Instead, they were swamped by a huge outpouring of votes in Republican-leaning areas of battleground states, particularly rural and small-town counties in Florida and the Midwest.

"We had [vote] goals that we set out that we thought were very realistic, that we thought could achieve victory," Devine said. "But a lot of people in rural areas participated in this process at levels that we have not seen before."

Another Kerry strategist said the campaign may have miscalculated the power of incumbency, especially during a time of heightened concern about terrorism. "It's easy to underestimate the reluctance in general that the American public would have in throwing out an incumbent president," the strategist said. "It's even more of a challenge when the country's perceived to be in some level of a war. That was an overriding backdrop that some of us tended to underestimate."

The Kerry camp also may have misjudged the power of Bush's appeal to social and cultural conservatives, even though White House senior adviser Karl Rove had explicitly set about to expand turnout among Christian conservatives.

Led by Rove, campaign manager Ken Mehlman, chief strategist Matthew Dowd and others, Bush's reelection team ran a disciplined operation that rarely deviated from the plan that was set from the start. Bush paid tribute to his team in his remarks yesterday, describing Rove, who has been at his side as he ascended through the Texas governorship to the presidency and now to a second term, as "the chief architect."

Bush's advisers, often second-guessed over their strategic decisions, took satisfaction not only from the victory but from the size of Bush's margin, which they said would end questions of legitimacy that had dogged him after 2000. Dowd, in a final strategy memo before returning to Texas, said the president had won more votes -- more than 59 million -- than any other candidate in history and that the campaign had succeeded in changing the shape of the electorate, raising Republicans to parity with Democrats.

"The other side did a very good job identifying their voters and getting them out to vote," Devine said. "It's just that simple."

Research editor Lucy Shackelford and political researcher Brian Faler contributed to this report.

President Bush appears at the Ronald Reagan Building with first lady Laura Bush and their daughters, Jenna and Barbara, right, and Vice President Cheney

and his wife, Lynne V. Cheney. In his victory speech, the president vowed to challenger John F. Kerry's supporters, "I will do all I can do to deserve your trust."Sen. John Edwards greets Sen. John F. Kerry at Faneuil Hall in Boston before Kerry's concession speech.First lady Laura Bush embraces her husband as daughter Jenna applauds.Sen. John Edwards and his wife, Elizabeth, listen as Sen. John F. Kerry gives his concession speech at Boston's Faneuil Hall.An emotional Teresa Heinz Kerry greets supporters of her husband in Boston.