Conservative Republicans and wary Democrats fear that Mr Bush might indeed try to steal the Democrats' clothes. "They talked about issues that people care about, and they won," he told the senators.

As a lame-duck president after years of poor relations with Democrats who were bitter about being frozen out of the decision-making process, turning opportunity into legislative reality will be one of the biggest challenges of Mr Bush's political career.

The message from the voters was that they rejected one-party rule in Washington and wanted to see whether divided government could lead to the kind of results Mr Bush was unable to achieve when Republicans were the kings of Capitol Hill.

Having been swept into power on the backs of their condemnation of a "do-nothing Congress", there will be pressure on Democrats to demonstrate that they can do business with Mr Bush. The president will want to leave office with achievements under his belt and use his power of veto sparingly.

"The Democrats should adopt a good government strategy rather than a take-no-prisoners strategy," said Senator Birch Bayh, a Democrat who served in the Senate for 18 years.

He believed that Republicans in Congress would adopt a new approach. "They've gotten the signal that people didn't like what was going on, that the well was poisoned. They're not going to be bomb throwers."

In 1996, President Bill Clinton and a Republican-controlled Senate and House of Representatives introduced a welfare reform Bill that became a landmark piece of social legislation. Both parties claimed credit for it. Republican presidents have also signed legislation sent to them by Democratic houses of Congress.

To the dismay of conservative Republicans, Mr Bush has already indicated that he agrees with the Democratic proposal to raise the minimum wage. But the centrepiece of his last 24 months in office could be a comprehensive overhaul of America's immigration laws.

His desire for tough border security measures combined with opportunities for many of America's 12 million illegal immigrants to become citizens is shared by more Democrats than Republicans on Capitol Hill.

Gardner Peckham, a senior aide to Newt Gingrich, speaker of the House after the Republican takeover of Capitol Hill in 1994, said: "Democrats will have to be very careful not to appear like they're looking backwards and wanting retribution.

"They won't want to look like they're the source of the problem. But they're not going to want the president to look good on any of these issues. The prize is the White House in 2008. Control of Congress is great, but without the White House you can't control the agenda."

Immigration, he suggested, would be "an interesting test" of whether genuine cooperation was possible.