To reach the compromise, the administration said Tuesday it had won agreement from the commission that it would seek no further public testimony from White House officials and that Rice's appearance would not be viewed as a precedent.
The commission welcomed the decision in a statement which said, "We will work with the White House to schedule both sessions promptly."
President Bush and Cheney have agreed to a single joint private session with all 10 commissioners, with one commission staff member present to take notes of the session, White House counsel Alberto Gonzales said in a letter to the panel. Previously, the administration was only offering private interviews of Mr. Bush and Cheney with just the commission chairman and vice chairman.
CBS News Correspondent Bob Fuss reports the decision came as Democrats were blasting the administration on the Senate floor. The Los Angeles Times reports Sens. Ted Kennedy of Massachusetts and Charles Schumer of New York planned on Tuesday to introduce a resolution calling on Rice to testify.
The decision to have Rice testify was made in the wake of the publication of former White House counterterrorism chief Richard Clarke's book (published by a company that, like CBSNews.com is owned by Viacom), in which he charges that the Bush administration. He also has about Rice.
The commission's Republican chairman, former New Jersey Gov. Thomas Kean, welcomed the decision and said the White House shouldn't be concerned that the testimony would violate the principles of executive privilege or separation of powers.
"We recognize the fact that this is an extraordinary event. There are all sorts of reasons why this is different. This does not set a precedent," he told reporters. He said there was still no time set — either for Rice's public testimony or for Mr. Bush and Cheney's private appearance.
The decision to have Rice testify follows the publication of former White House counterterrorism chief Richard Clarke's book, in which he charges that the Bush administration was slow to act against the threat of al Qaeda.
Clarke testified before the commission last week that the Bush administration did not make fighting al Qaeda an urgent priority before the 2001 attacks and was fixated on Iraq. The White House has denied his allegations.
Riceto criticism by Clarke that President Clinton "did something, and President Bush did nothing" before Sept. 11 and that both "deserve a failing grade."
Rice responded: "I don't know what a sense of urgency — any greater than the one that we had — would have caused us to do differently."
Rice had previously offered to meet with the commission in private for a second time. But the heads of the commission indicated on Monday that if she refused a public session, any private interview would be conducted under oath and a transcript made.
The Los Angeles Times reports no transcript was made of Rice's first, four-hour meeting with the commission, although some panel members took notes. And according to the Washington Post, only six of the 10 commissioners attended that session because it was held on a Saturday at the White House.
In a 60 Minutes interview aired Sunday, Rice cast herself as ready to testify but restrained by the principle of executive privilege.
"Nothing would be better from my point of view, than to be able to testify. I would really like to do that. But there's an important principle involved here," Rice said "It is a long-standing principle that sitting national security advisors do not testify before the Congress."
Regarding Rice's upcoming public testimony and the charges in Clarke's book, "There are significant and extremely important questions," said commissioner Tim Roemer, a former Democratic congressman.
Roemer said those questions include, "Was there an urgent priority in the Bush administration in fighting terrorism? How quickly was the decision made? Was it slow as Mr. Clarke says or accelerated as Dr. Rice says?"
Commissioner Slade Gorton, a former Republican senator, said the Sept. 11 panel accepted the proposal in a meeting Tuesday, including the stipulation that it not call other White House officials because "we hadn't planned to."
"I think the White House would have been better off if it had made the agreements sooner, but I'm delighted," said Gorton. "I have felt all along that her public testimony would be good for the country."
Commissioner Bob Kerrey, a former Democratic senator, said the president and vice president will not be under oath in their meeting with the commission. Kerry said Rice's testimony will be critical in determining what the Bush administration could have done to prevent the attacks and that the White House "made the right decision."
Aboard Air Force One, White House spokesman Scott McClellan said Mr. Bush had decided over the weekend to pursue an arrangement with the commission for Rice's testimony "provided that we can uphold this important principle. It's important to protect the principle of separation of powers between the executive and legislative branch." The president was to make public remarks about the agreement later Tuesday.
The session with the president and vice president would be without set time constraints, Gorton said. Originally, Mr. Bush and Cheney had wanted to restrict any meeting to one hour, although Mr. Bush relaxed that requirement earlier this month. Former President Bill Clinton and former Vice President Al Gore agreed last month to private meetings without restrictions. Regarding Rice, Gorton said a previous proposal to make public notes from her private meeting with the panel in February is now moot and won't be done.