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The White House also said in the statement that Washington respects the "vote of the Iranian people", while congratulating them for "their participation in the political process."
"It is our hope that the Iranian government will heed the will of the Iranian people and make responsible choices that create a better future for all Iranians," said the White House.
The United States has long accused Tehran of secretly developing nuclear weapons, but Iran has denied the accusation, insisting its nuclear program is meant for civilian purposes only.
On Saturday, French Foreign Minister Laurent Fabius said that his country is ready to work with the Iranian incoming president, Rouhani, on a range of issues including the nuclear issue of the country and the Syria conflict.
Fabius acknowledged that the recent election was in line with the Iranians "desire for democracy."
Also, German Foreign Minister Guido Westerwelle welcomed the Iranians' choice as a vote for change and "a constructive foreign policy."
In a message after winning the election, Rouhani promised to form a government of "wisdom and hope" and draw up a blueprint on civil rights.
Analysts attributed his win to a large turnout of voters who saw him as the best choice to translate their hopes into real changes in Iran's domestic and foreign policies.