On the heels of a tete-a-tete with Barack Obama, Hillary Clinton has set the time and place for her formal exit from the Democratic primary race.

The New York senator plans to drop out of the race at noon Saturday, at the National Building Museum in Washington, D.C.. The announcement came Friday after some initial confusion over when and where Clinton would concede.

With the details in place, party leaders hope the rally -- where Clinton plans to endorse Obama -- will serve as an effective call for party unity.

The two Democrats apparently began mending fences when they met in private Thursday night.

California Sen. Dianne Feinstein, a major Clinton supporter who hosted the meeting at her Washington, D.C., home, said the two emerged from the hour-long session "laughing."

Most of what Obama and Clinton talked about stayed secret in the hours after their meeting, and only a short, cryptic campaign message gave any hint of what happened behind closed doors.

"Senator Clinton and Senator Obama met tonight and had a productive discussion about the important work that needs to be done to succeed in November," their campaigns said in a joint statement released late in the evening.

But Clinton supporters swiftly kicked up speculation that she's being considered as Obama's running mate, just hours after the campaign declared she's not seeking the No. 2 spot.

Feinstein was promoting the "dream ticket" concept the next morning.

"She brings constituencies ... like women, like elderly, like working people, certain states that are critical to win. I also think it's a very good mix because it's a very historic ticket," Feinstein told FOX News Friday when asked about the upsides of an Obama-Clinton pairing.

Feinstein said she set up Clinton and Obama Thursday night by her fireplace, and then went upstairs to give them privacy. She said she originally offered up her home to the candidates for the meeting, and Clinton took her up on the offer.

VoteBoth.com, a Web site founded by former Clinton aides and dedicated to uniting the rivals, said, "We are thrilled to hear that Senators Obama and Clinton are meeting already. We hope that Thursday night's conversation is a step toward an Obama-Clinton ticket that will unify the party."

New York Sen. Charles Schumer, a top Clinton supporter, said Friday that the meeting was not about the vice presidency, but that Clinton would accept the No. 2 spot.

"She has said if Senator Obama should want her to be vice president and thinks it would be best for the ticket, she will serve, she will accept that. But on the other hand, if he chooses someone else she will work just as hard for the party in November," Schumer told ABC's "Good Morning America."

Party leaders hope the Saturday Clinton rally will help heal the lingering animus among Democrats.

Clinton said in an e-mail to supporters that: "On Saturday, I will extend my congratulations to Senator Obama and my support for his candidacy. This has been a long and hard-fought campaign. ... I have said throughout the campaign that I would strongly support Senator Obama if he were the Democratic Party's nominee, and I intend to deliver on that promise."

Many Clinton supporters think the best way to unite the party is to unite the candidates.

But in a statement Thursday evening, the Clinton campaign stressed that the running-mate decision is up to Obama, and that Clinton "is not seeking the vice presidency."

Obama has indicated he intends to take his time making a decision. On Wednesday, he named a three-person vice presidential vetting team that included Caroline Kennedy, daughter of the late President Kennedy.

FOX News' Major Garrett and Trish Turner and The Associated Press contributed to this report.