Justice Rehnquist, who had been suffering from thyroid cancer, died at his home in the Virginia suburbs late on Saturday night. Although his illness kept him off the bench for several months earlier this year, he did not resign. Indeed, despite his visible frailty, he insisted on swearing in Mr Bush as President in January.
In a brief appearance at the White House yesterday, Mr Bush promised to name a new chief justice in "prompt" and "timely" fashion. He paid warm tribute to Mr Rehnquist, praising his "powerful intellect" and "commitment to the rule of law". His death was "a great loss for our country," the President said.
Mr Rehnquist, who was 80, was appointed by Rich-ard Nixon and served on the court for 33 years, the last 19 of them as chief justice. He was a conservative, but of the pragmatic, rather than ideological variety.
During his long tenure as the 16th chief justice, he emerged as a masterful manager of one of the most important institutions in the US, a referee of last resort for the social and cultural wars that divide the country, from abortion to gay rights and the respective roles of church and state.
He won plaudits for the low-key way in which he presided over impeachment proceedings against President Clinton in January 1999. But the most important single vote of his career came in December 2000, when Mr Rehnquist led the 5-4 majority by which a bitterly divided court cut off the election recount in Florida and gave Mr Bush the White House.
The President now faces a highly complicated and delicate choice - when he is already under furious criticism for his handling of the Katrina situation. Even before Justice Rehnquist's death, there was already one vacancy on the court, following the resignation this summer of Sandra Day O'Connor, regarded as its critical swing vote.
To replace her, Mr Bush nominated the federal appeals judge John Roberts, whose senate confirmation hearings were due to start tomorrow, but which might be postponed because of Katrina. They could also be affected by the new vacancy left by Justice Rehnquist.
In essence, Mr Bush must decide whether to elevate one of the sitting justices - staunch conservatives Antonin Scalia and Clarence Thomas are the most obvious candidates - to the top job, or to nominate an outsider.
Mr Rehnquist himself was already on the court when he was appointed chief justice in 1986. But if Mr Bush followed Ronald Reagan's course then, the upshot would be not one, but two sets of time-consuming and probably tendentious hearings on Capitol Hill. Even if he is already on the court, a new chief justice must be confirmed in that post separately from the hearings to approve a successor as one of the eight associate justices.
Leading non-court candidates include Alberto Gonzales, the attorney general, a long-standing friend of the President who would be the first Hispanic chief justice if confirmed. Michael Luttig and Harvie Wilkinson, two conservative federal appeals court justices, are also in the running, along with Theodore Olson, solicitor-general during Mr Bush's first administration.
But, whatever happens, a new chief justice is unlikely to be in place before November. Every sign is that Mr Roberts, who has impressed even many Democrats on Capitol Hill, would be comfortably confirmed. But even then the court would only have eight members.
Some suggest that to save time, Mr Bush could renominate Mr Roberts as chief justice. Or he could ask Ms O'Connor to become chief justice for a short period.