The Taoiseach, Mr Ahern, said he wished President Bush every success and looked forward to working closely with him "on the whole range of common interests between our two countries, political, economic and cultural".
Mr Ahern said the President had written to him last week confirming his strong personal interest and ongoing commitment to the peace process.
"I look forward to working with him and to continuing the strong tradition of friendship and co-operation that has characterised relations between our countries in the past," the Taoiseach said.
Congratulations to the newly inaugurated Mr Bush continued to pour in from world leaders, united in the desire to keep the new administration engaged on the international stage.
In Rome, Pope John Paul II called on Mr Bush to build "a society marked by true justice and freedom", in a telegram wishing him well at the start of his term. He said he hoped that under the new leadership the American people would "rediscover, among its rich religious and political traditions, the spiritual values [that] lay the ethical foundation for the construction of a society marked by true justice and freedom".
The British Foreign Secretary, Mr Robin Cook, praised Mr Bush's foreign policy team, singling out the new US Secretary of State, Gen Colin Powell.
Germany's Chancellor, Mr Gerhard Schroder, stressed the links with the US in an official letter of congratulation in which he pointed to their joint role at the heart of the NATO alliance.
The French President, Mr Jacques Chirac, said in a message that the United States and France had to face up to great challenges present in the world. "They also have the highest responsibilities to assume on the international stage, notably at the United Nations."
In the Middle East, the Iranian Foreign Ministry spokesman, Mr Hamid-Reza Asefi, called on Washington to change its policy towards Iran. But a report on state television said Mr Bush's inaugural address had implied that "Washington's policy of arrogance, domination and hegemony" would continue.
The Palestinian Liberation Organisation said in a statement it was confident the Bush administration would play "a positive and balanced role in pushing forward the peace process."
There was no message from China, however, where Mr Bush's plan to press ahead with a more aggressive defence agenda has been condemned. The official press made no mention of the inauguration.
Europe's relationship with the Bush administration could get off to a bad start over the Balkans weapons controversy, writes Conor O'Clery. EU foreign ministers meeting in Brussels are facing a wave of public concern about the alleged health risks of depleted uranium shells and complaints by Germany that Washington kept its European allies in the dark. The Pentagon has twice sent US army medical experts to NATO headquarters to help to reassure the European media.