"The secretary of state is America's face to the world, and in Dr Rice the world will see the strength, the grace and the decency of our country," Bush said with Rice - but not Powell - at his side in the White House on Tuesday.
"During the last four years I've relied on her counsel, benefited from her great experience, and appreciated her sound and steady judgment. And now I'm honoured that she has agreed to serve in my cabinet," he said.
The president also announced that Rice's deputy, Stephen Hadley, would replace her as national security adviser, a post which, unlike secretary of state, does not require confirmation by the US Senate.
"Steve is a man of wisdom and good judgment. He has earned my trust. And I look forward to his continued vital service on my national security team," Bush said in prepared remarks in the Roosevelt Room of the White House.
Powell's resignation was announced earlier on Monday.
Powell was not present when
Bush made the announcement
Rice has been national security adviser since Bush took office in January 2001.
Of his top aides, she spends more time with the president than any other except White House Chief of Staff Andrew Card. She was at his side at all times in the run-up to the Iraq war.
In April, she gave dramatic testimony to the commission investigating the 11 September 2001 attacks, in which she rejected criticism that the Bush administration was negligent in the summer before the attacks and said there was no "silver bullet" that could have averted them.
She was frequently at Bush's side on the campaign trail, and often travels with the Bush family to the president's Camp David retreat on weekends.
Defence Secretary's post
Rice had been talked about as a possible successor to Defence Secretary Donald Rumsfeld, and that job was believed to have been her preference, but there has been no indication that Rumsfeld is leaving any time soon.
Rice, who turned 50 on Sunday, was previously the provost of Stanford University and has debated whether to quit her high-stress Washington life and return there. But many of her aides believe if Bush told her that he needed her to stay, she would do so.
Rice had been seen as a possible
successor to Donald Rumsfeld
Rice's training is in Russian affairs, but in her current job she has focused on all the world's hot spots.
She met Palestinian Prime Minister Ahmad Quraya in Berlin in May in what was the highest-level US contact with Quraya at the time.
In mid-July she travelled to Asia for meetings with officials in Japan, South Korea and China.
Prior to being Rice's deputy, Hadley had served as a senior foreign and defence policy adviser to Bush during the presidential campaign, and worked in the transition on the National Security Council.
Hadley last year accepted blame for a reference to Iraq seeking uranium in Niger that showed up in Bush's January 2003 State of the Union speech and sparked a controversy after it was shown to be based on forged documents.
Stephen Hadley will replace Rice
as national security adviser
George Tenet, then director of the CIA, also took responsibility for the inclusion of the false accusation in the address.
Hadley served as the assistant secretary of defence for international security policy from 1989-1993 and was a partner in a Washington DC law firm.