Chief Justice Rehnquist's death comes two months after Justice Sandra Day O'Connor announced her intention to retire from the high court.
The president has nominated Judge John Roberts to take her place, and the Senate is expected to vote on whether to confirm his nomination sometime before the beginning of the next Supreme Court session on October 3rd.
President Bush says he will move quickly to nominate a successor to Chief Justice Rehnquist.
"He was extremely well respected for his powerful intellect," Mr. Bush. "He was respected for his deep commitment to the rule of law and his profound devotion to duty. I will choose in a timely manner a highly qualified nominee to succeed Chief Justice Rehnquist."
The president could choose to elevate one of the other associate justices to the position of chief justice of the nine-member court. But that person would require separate confirmation hearings before the Senate Judiciary Committee followed by a full Senate confirmation vote. The president would then have to select someone to replace the associate Supreme Court justice who is being elevated.
The last time there were two vacancies on the Supreme Court was in 1971, and William Rehnquist was chosen by President Richard Nixon to fill one of the openings.
Democrats are urging President Bush to consult with them about his next pick, much as he did before he nominated Judge John Roberts to succeed Justice O'Connor.
Senator Chris Dodd is a Connecticut Democrat who spoke on Fox television.
"He (Bush) invited the (congressional) leadership down to (including) the (Senate) Judiciary Committee, the Senate," Senator Dodd says. "There was at least some conversation about that consultative process, the advise and consent role in the Constitution. And I would strongly recommend, however he decides to proceed over the coming months, that he continue that process."
Even as the president and Senate prepare for the process of filling two Supreme Court vacancies, Chief Justice Rehnquist is being honored for his long service on the high court and for his legacy.
After his initial appointment as an associate justice in 1971, William Rehnquist was elevated to the post of chief justice in 1986 by President Ronald Reagan.
He presided over the impeachment trial of President Bill Clinton before the Senate in 1999, and voted with the court majority in 2000 to stop the vote recounts in Florida that made George W. Bush president over Democrat Al Gore.
In his early days on the Supreme Court, William Rehnquist was known as the Lone Ranger for his lonely conservative stands on a range of issues.
But that began to change as the make-up of the court became more conservative in the 1970's and 1980's.
Charles Fried is a professor of constitutional law at Harvard University in Massachusetts.
"So, what you had was a radical shift to the extreme left in the 1960's, and Rehnquist just really hauling it back to the middle. And that is the great legacy, because many of the important things have stuck," Mr. Fried says. "The death penalty is there (legal) but it is under very rigorous scrutiny."
Experts say Chief Justice Rehnquist will also be remembered for leading a gradual shift on the court that increasingly favored the rights of states over the central government on a wide range of issues.
Stephen Wermiel is a longtime expert on the Supreme Court at the American University's Washington College of Law.
"That, I think, probably will be his most significant legacy," Mr. Wermiel says. "That we had been in a period of about 60 years since the New Deal (economic program under President Franklin Roosevelt), in which the court had not really second-guessed (questioned) the power of Congress at all. I mean never. And, in the last decade, the court has been regularly second-guessing the power of Congress.
Chief Justice Rehnquist is also being remembered for the dedication and leadership he demonstrated in his 33 years of service on the high court. Though diagnosed with thyroid cancer last October, he preferred to work right up until his passing.
He once reflected on the public perception of the Supreme Court during his tenure in an interview with the CSPAN public affairs network.
"I think the court generally has a pretty good image," Chief Justice Rehnquist said. "I think most people think of it as doing its work reasonably well, though they understand it very imperfectly. And, I think, particularly, compared to the other branches of government, it probably does not suffer by comparison with them."
Until a new chief justice is confirmed, the court will be led by the senior associate justice, Justice John Paul Stevens. He is 85-years old, and was appointed to the high court by President Gerald Ford in 1975.