The European Commission President, Mr Romano Prodi, and the Dutch EU Presidency have congratulated President Bush on his victory in the US election.
"I hope his second term will see the United States and the whole world enjoy the political stabilisation and the guarantee of collective security we have all done our utmost to ensure," Mr Prodi said.
The outgoing Commission president reflected European concerns about a second Bush term, however, when he stressed the importance of a multilateral approach to international relations.
"Europe will continue to work to strengthen its bonds of friendship and co-operation with the US. Those bonds, which have never been called into question, are vital to maintaining peace in the world on the basis of multilaterally shared principles and values," he said.
The Dutch Prime Minister, Mr Jan Peter Balkenende, said he looked forward to "a continuation of the productive relations" between Europe and the Bush administration.
"The United States and the European Union are linked by strong cultural, economic and political ties, and by our shared values. This makes us each other's natural and indispensable partners.
The presidency is confident that this relationship will be further deepened and strengthened under the new administration," he said.
Such warm sentiments disguised a bleak mood in Brussels yesterday as the outcome of the US election became clear.
Many EU politicians and diplomats hold Mr Bush personally responsible for the divisions within Europe caused by the Iraq war and for a new, harsh tone in Washington's approach to its allies.
Some optimists yesterday expressed the hope that, in his second term, Mr Bush would adopt a more co-operative approach towards America's allies and working with Europeans as equals.
The US ambassador to the EU, Mr Rockwell Schnabel, yesterday promised "a renewed commitment to reach out to Europe", adding that Washington was eager to listen to European concerns.
"You're going to see a United States that is interested in listening to your concerns, that is going to be dealing with you, coming from humility and coming from respect."
Green MEP Mr Daniel Cohn-Bendit was unimpressed by such blandishments, declaring that any change in US foreign, security, environmental and social policy was now improbable. For Mr Cohn-Bendit, the proper response to a more conservative America is to create a stronger Europe.
"In the next four years the role of the European Union as a counterweight to the US will become ever more important. Only a strong, united Union that is able to act decisively when needed, will be capable of fulfilling this task. Therefore, a rapid ratification of the Constitution must become priority number one in Europe. If we fail to adopt our Constitution, the second instalment of the Bush administration will be popping corks in the White House," he said.
Europe's difficulty is that it needs US co-operation to fulfil many of its own foreign policy aims, notably a resolution of the conflict between Israelis and Palestinians. Some EU policy-makers blame particular voting blocs in the US for Mr Bush's reluctance to engage with the Middle East peace process but others fear that the removal of electoral pressure during a second term will have little impact on his approach to the region.
Good relations between the EU and Washington are also essential to the success of arrangements for co-operation between NATO and the EU's emerging defence identity.
The election of a number of deeply conservative new members of Congress and the importance of "moral values" - opposition to gay rights and abortion - to Mr Bush's victory has further deepened the sense of estrangement between policy makers on both sides of the Atlantic.
It will be the task of the former taoiseach, Mr John Bruton, to build new networks of communication between policy elites in the EU and the US when he takes up his post as EU ambassador to Washington later this month. Mr Bruton will also seek to explain the EU to a sceptical American public and to help EU politicians to understand the concerns of their US counterparts.
While many in Brussels were yesterday contemplating the next four years with dull despair, a few saw in the US election outcome an opportunity to assert Europe's independence from an ally that no longer shares the same interests and values.