Illinois Sen. Barack Obama was widely embraced as the presumptive Democratic presidential nominee across Capitol Hill on Wednesday, as his rival, Sen. Hillary Rodham Clinton (N.Y.), plotted an exit strategy.

Democrats on both sides of the Capitol expressed a mixed sense of elation at Obama’s historic achievement, relief that the protracted contest had reached its conclusion and deference to Clinton, who did not concede defeat after it was clear Obama had crossed the delegate threshold on Tuesday night.

Party leaders and Members turned immediately to urging unity and expressed their belief that the fallout from the nomination fight will not hamper their effort to reclaim the White House in the fall.

“We’re going to see one party come together,” said Rep. Gregory Meeks (D-N.Y.), a Clinton backer and member of the Congressional Black Caucus.

Speaker Nancy Pelosi (D-Calif.), who has remained publicly neutral in the contest, praised Clinton but acknowledged the race was done and urged the party to unify behind Obama.

“The people have spoken. The elections are over for the nomination. Barack Obama is the nominee of the Democratic party,” Pelosi said.

Pelosi talked to Obama on Wednesday at the meeting of the American Israel Public Affairs Committee.

While both Clinton and Obama were in Washington, D.C., on Wednesday to speak to AIPAC, only Obama returned to the Senate chamber.

A group of previously neutral Senators, after meeting twice this week, released a letter Wednesday calling for party unity.

“Our focus now is on victory in November and on giving Barack Obama every ounce of our support, every bit of our energy, and our total commitment to do everything in our power to win the presidency,” the Democrats wrote. “We want to say, to the Clinton supporters who worked their hearts out and who we know are very disappointed, that their extraordinary effort on her behalf has strengthened our party and strengthened our nation.”

Longer-term Obama backers took a moment to savor the long-fought and groundbreaking victory.

“Sen. Obama’s achievement is historic,” said a euphoric Rep. William Lacy Clay (D-Mo.), an Obama backer. “I am so proud of my friend from Chicago, Barack Obama, of what he’s been able to accomplish.”

Obama will campaign this morning in Bristol, Va., where he will be welcomed by Rep. Rick Boucher (D-Va.), who endorsed the Illinois Senator in January.

Obama was forced to cancel a rally in southwestern Virginia before the state’s February primary, and Boucher said he had been promised Obama would make up for it as soon as he could.

The 3,000 tickets for the rally were gone in a few hours. Boucher said that he believed it was the first time since 1960 that a major-party presidential nominee has campaigned in southwestern Virginia and that he remembered being at that rally with John F. Kennedy as a teen.

While Obama won a convincing victory in Virginia’s February primary, Clinton handily carried Boucher’s rural, largely white, economically depressed district. Boucher said Wednesday that he was confident that Obama would make inroads with his constituents and other rural voters, who will be “drawn to him” after they get to know him.

Meanwhile, most Members said Clinton should be given time to figure out her strategy for conceding the race, but they were split about whether Clinton should be offered the vice president slot.

“I understand what the Clintons are going through. They have to realize he’s the nominee,” Clay said. “It will probably take a few days for that to sink in.”

Clinton gave clear signals Tuesday night that she doesn’t intend to continue to campaign to the convention but asked for time to hear from her supporters about what her next move should be.

House Majority Whip James Clyburn (D-S.C.) said that Clinton should concede quickly, but that her delayed capitulation would not harm Democrats.

“It doesn’t hurt me, and it doesn’t hurt the party. It may hurt her,” he said.

Rep. Debbie Wasserman Schultz (D-Fla.), a national campaign co-chair for Clinton, urged the Obama campaign to put Clinton on the ticket and called for patience from people demanding that she concede.

“There needs to be the requisite amount of patience,” she said. “There is an understanding of the narrowness of time that she has. There is no one who is more acutely aware of that than her.”

Talk of Clinton as a possible running mate for Obama ratcheted up Tuesday after she told colleagues she would be open to the idea.

“I think she should be on the ticket,” said Rep. Jim McGovern (D-Mass.), one of a handful of Clinton supporters who joined her on stage after her Baruch College speech on Tuesday night. McGovern noted that while the decision rests with Obama, both candidates won roughly 18 million votes. “Why don’t we bring them together and focus on winning in November?”

Other Clinton advocates said the decision rests with Obama alone.

“You don’t earn a spot on the ticket,” said Rep. Barney Frank (D-Mass.). “It’s not a merit badge.”

Rep. Jim Cooper (D-Tenn.), a longtime Obama backer, said he isn’t a fan of putting Clinton on the ticket.

“You don’t want a competitor on the ticket, and you don’t want Bill wandering around the White House,” Cooper said. “Have you read this Vanity Fair article? Amazing.”

Early Wednesday, Pelosi and Senate Majority Leader Harry Reid (D-Nev.) released a letter calling on all remaining uncommitted superdelegates to make their preferences known by the end of Friday.

Majority Leader Steny Hoyer (D-Md.) stopped short of making an official endorsement, but he said he is “enthusiastic about an Obama candidacy.”

House Democratic Caucus Chairman Rahm Emanuel (Ill.), a former aide in Bill Clinton’s White House who had a delicate balancing act as a superdelegate, formally came out Wednesday for Obama, as did Democratic Congressional Campaign Committee Chairman Chris Van Hollen (Md.).

Still, some Members have no intention of formally weighing in on the presidential contest; many of the roughly three dozen House Members who remain undeclared hail from conservative districts where the majority of voters aren’t likely to support the Democratic presidential candidate in November.

“The Speaker and I don’t agree on everything,” said Rep. Gene Taylor (D-Miss.), adding that he doesn’t intend to make an endorsement.

Others, such as Reps. Dan Boren (D-Okla.) and Charlie Wilson (D-Ohio), said they see little reason to weigh in now because the contest is essentially over.

“I think the system has worked,” Wilson said.

In addition to the group of Senators who broke their silence Wednesday, Montana Democrats Sens. Max Baucus and Jon Tester endorsed Obama late Tuesday night after their state’s balloting concluded.

This election year has been particularly difficult for Senators because two of their own were battling for the Democratic nomination. Many didn’t want to pick sides, including Reid, who has tried to maintain impartiality in the race between two of his colleagues. Reid finally said Tuesday that he would endorse no later than Friday.

For the most part, Democratic Senators have wanted to give Clinton space and time to determine how best to end her campaign. Party leaders didn’t want to weigh in until the final primary votes were cast in Montana and South Dakota on Tuesday and until after she decided her next move.

With that sentiment in mind, Sen. Claire McCaskill (D-Mo.), an early supporter of Obama’s campaign, said Clinton should be given that deference but cautioned Clinton against withholding her support from her one-time rival.

“If she were not to come forward and be a supporter of Sen. Obama that would be a problem, but I don’t think that is going to happen,” McCaskill said. “I think she is going to do the right thing because she’s a good person and she cares more about America than she does [about] herself.”

Steven T. Dennis, Tory Newmyer and Tim Taylor contributed to this report.