The illness of Chief Justice William Rehnquist opens up the prospect that President Bush might in his second term exercise one of the key powers a US president has - to nominate Supreme Court justices.
By Paul Reynolds
World Affairs correspondent, BBC News website
The nine Supreme Court justices are appointed for life
It was something he could not do in his first four years. There were no vacancies.
But now there might be. And Mr Bush could have an opportunity to add the Supreme Court to the spheres of influence the Republicans exercise over the presidency and both houses of congress.
There are few things a president likes doing more.
By getting a like-minded nominee on to the nine-member court, a president can leave a legacy that goes on working for years.
The justices are in effect politicians in black robes. And they are appointed for life.
At the moment the court is fairly evenly split between liberals and conservatives and quite often the decisions go 5-4 one way or another.
In the case that cleared the way for George W Bush himself to win the election in 2000, the decision was itself taken 5-4.
There has not been a new justice on the court for more than 10 years, so Mr Bush might have an opportunity to strengthen the conservative element.
Chief Justice Rehnquist, a conservative appointed by President Nixon in 1972, has thyroid cancer and is aged 80.
Sandra Day O'Connor, appointed by President Reagan, is 74. The leading liberal, Justice John Paul Stevens, is 84. Most of the justices are over 65.
Mr Bush himself has indicated that he will look for justices in the conservative mould of his two favourites on the court - Justices Antonin Scalia and Clarence Thomas, the only African American on the bench.
It does not always quite work out as a president plans.
Mr Bush's father appointed Justice David Souter, thinking that he was reliably conservative. He proved not to be and in fact he was one of the minority who wanted a recount in Florida in 2000.
There is one hurdle for the president to overcome - the Senate, which has to confirm any nominee.
This often leads to fierce battles, as everyone knows that future public policy is at stake.
The Republicans have a majority in the Senate but not a big enough majority to stop a filibuster, or delaying tactics, by the Democrats.
The justices ruled Guantanamo prisoners should have court access
It is on such occasions that the kind of talk about unity and coming together, which Mr Bush mentioned in his acceptance speech, is largely forgotten.
The importance of the Supreme Court in US public life cannot be overstated.
It makes and breaks laws.
This is something rather difficult for many from a parliamentary background to accept.
But in the American federal system, all laws have to conform to the Constitution and the Supreme Court is the judge.
Recently, it intervened in the detention of prisoners held at Guantanamo Bay, saying that they should have access to the US courts.
That was an example of how the conservative/liberal balance can swing towards the liberal.
Two of its most famous decisions changed policy on racial segregation and abortion.
In 1954, it ruled on a case brought by a man named Oliver Brown whose seven-year-old daughter Linda had been denied a place at a whites-only school in Topeka, Kansas.
In that case, Brown v Board of Education, it struck down a ruling it had made itself in 1896 that "separate but equal" facilities were legal.
In 1973 an unmarried pregnant woman named only as Jane Roe complained that she had been unlawfully denied an abortion in Texas.
In Roe v Wade, the Court agreed. It decided that the right to an abortion was implied by the established principle of a right to privacy.
This case, Roe v Wade, is one that the anti-abortion movement in the United States wants to overturn.
And that is why nominations for the Supreme Court are watched so closely.