White House press secretary Scott McClellan, on Air Force One with President Bush, said the commission had unanimously agreed to the conditions.
President Bush and Vice President Dick Cheney also sweetened their previous offer to the commission. They will still testify in private, but to the whole commission, not just the chairman and vice-chairman, reports CBS News Correspondent Dan Raviv.
"The president has consistently stated a policy of strong support for the commission and instructed the executive branch to provide unprecedented and extraordinary access to the commission," the commission said in a statement. "His decisions today reflect that policy of strong support, and we welcome them."
The decision to let Rice testify was conditioned on the Bush administration receiving assurances in writing from the commission that such a step does not set a precedent and that the commission does not request "additional public testimony from any White House official, including Dr. Rice," White House counsel Alberto Gonzales said in a letter to the panel.
Subject to the conditions, the president will agree "to the commission's request for Dr. Rice to testify publicly regarding matters within the commission's statutory mandate," Gonzales's letter stated.
"The president recognizes the truly unique and extraordinary circumstances underlying the commission's responsibility to prepare a detailed report on the facts," Gonzales added.
Congressional leaders, Gonzales noted, have already stated that this would not be a new precedent.
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 is 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.
Clarke testified before the commission last week. He testified 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."
In resisting calls to let Rice testify in public, Mr. Bush had said he did not want to compromise his ability to keep private the advice he receives from aides
But there was intense political pressure to allow Rice to appear. On Monday, the change in policy was being discussed at the highest levels in the White House, reports CBS News Senior White House Correspondent John Roberts.
After the parade of current and former officials that have gone before the commission, Rice's absence, say former White House officials, created the perception that she has something to hide.
The White House also faced charges of hypocrisy — allowing Rice to appear frequently on television but not to testify, and claiming executive privilege at the same time it made public confidential e-mails and transcripts of background briefings to discredit chief critic Clarke.
Republican lawmakers have moved to declassify Clarke's earlier testimony in 2002, hoping to show discrepancies between Clarke's recent attacks on the administration's terrorism policies with flattering statements he made as a White House aide.
Clarke said he welcomed such a move, and also suggested they declassify all e-mails, memos and all other correspondence between him and Rice, as well as her private testimony before the commission.
The New York Times reports commission chairman Thomas Kean, a former Republican New Jersey governor, and Lee Hamilton, a retired Democratic congressman, say they are willing to declassify notes taken during Rice's earlier session.