Cheney named attorney David Addington as his chief of staff and John Hannah as his national security adviser.
Addington has been Cheney's counsel and Hannah has been his deputy national security adviser.
Libby resigned last week after he was indicted on five counts of obstruction of justice, perjury and making false statements relating to his testimony to the FBI and a grand jury investigating the leaking of CIA agent Valerie Plame's identity.
If convicted on all five counts, he could face as much as 30 years in prison and $1.25 million in fines. Libby is to be arraigned on Thursday, CBS News has learned.
The indictment has triggered calls on Capitol Hill for an apology from the White House and staff changes.
"There has not been an apology to the American people for this obvious problem in the White House," Senate Democratic leader Harry Reid said Sunday. He said President Bush and Cheney "should come clean with the American public."
the Special Counsel's statement (.pdf).
Reid, D-Nev., also said that Karl Rove, the president's closest political adviser, should step down. Rove has not been charged with a crime.
Reid added, "This has gotten way out of hand, and the American people deserve better than this."
However, White House press secretary Scott McClellan played down the idea of major changes on the president's staff, saying there had been no discussion of that beyond the usual vacancies that occur.
McClellan also brushed off Democrats' calls for Mr. Bush to apologize for the actions by administration officials, and for Rove to resign.
McClellan said Democrats are "trying to politicize the process." He said the White House won't comment while the case is before the courts.
"We don't want to do anything from here that could prejudice the opportunity for there to be a fair and impartial trial," presidential spokesman Scott McClellan said.
Meanwhile, Plame's husband, a former ambassador and outspoken critic of the war in Iraq, told
"It was the White House that decided they were going to go after me, and the tactic they decided to use was to go after my wife," said former ambassador Joseph Wilson.
Wilson also claimed that blowing his wife's cover put her in danger.
"There have been specific threats," he said. "Beyond that, I just can't go."
Mr. Bush and Cheney gave glowing endorsements and expressed no criticism of Libby after the senior White House adviser was indicted, resigned and lost his security clearance.
Cheney called Libby "one of the most capable and talented individuals I have ever known." Mr. Bush said Libby "has worked tirelessly on behalf of the American people and sacrificed much in the service to this country."
Reid said he was disappointed that Mr. Bush and Cheney expressed support for Libby in their public statements.
"The vice president issues this very terse statement praising Libby for all the great things he's done. Then we have the president come on camera a few minutes later calling him Scooter and what a great patriot he is," Reid told ABC's "This Week."
"I think they're as disappointed as I am ... almost dejected," Reid said.
A veteran Republican senator said that Mr. Bush needs to bring "new blood" into his White House.
Sen. Trent Lott, R-Miss., said the president should be on the lookout for "new blood, new energy, qualified staff, new people in administration." He said poor advice may have even contributed to the failed nomination of Harriet Miers to the Supreme Court.
The president's overall job approval was at 39 percent in an Associated Press-Ipsos survey conducted in early October. The poll also found that only 28 percent of respondents said the country was headed in the right direction.
Sens. Lindsey Graham, R-S.C., and Charles Schumer, D-N.Y., said the White House should conduct its own investigation of the CIA leak. Graham, however, said allegations of illegal activity appeared to be focused only on Libby.
"I think the likelihood of Karl Rove being indicted in the future is virtually zero," Graham said on CBS' "Face the Nation."
"I think this will be seen in history and in politics as Mr. Libby giving false information, if proven, and it will not be about an effort by the vice president to disclose a CIA operative."
The closest the indictment comes to Rove is its discussion of an unnamed senior White House official who talked to columnist Robert Novak about Plame and discussed the matter with Libby. That could describe Rove.
Special Counsel Patrick Fitzgerald said his investigation is "not quite done," but declined comment on Rove during a news conference on Friday.
Schumer said the investigation showed Cheney's office was in a campaign to discredit Wilson's wife because of his criticism of the administration's use prewar intelligence about Iraq's weapons of mass destruction and that Rove, despite public statements to the contrary, had discussed Wilson's wife with reporters.
If necessary, Schumer said, the president should take Cheney "to the woodshed."
"The president, again, ought to have some nonpolitical person look into this and see what should be done," Schumer said. "The standard shouldn't just be escaping indictment."