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Greetings from the Truman Library,
Thank you for your question! The president would often release a written statement and then verbalize parts of it for the benefit of the media and newsreels. This is the case for the president's statement of August 6, 1945. President Truman authorized the release of the full statement in Washington, DC, while he was traveling back from the Potsdam Conference on the USS Augusta. Then, aboard the Augusta, he verbalized portions of the statement for the news media. The most extensive audiovisual we can find at the Truman Library is available in the National Archives Catalog here: https://catalog.archives.gov/id/23630. This footage is in the public domain; no permission is needed to use this material. We kindly ask for credit if possible: 111-ADC-9865, US Army Signal Corps, National Archives and Records Administration.
Here is the text of what appears in this footage:
A short time ago an American airplane dropped one bomb on Hiroshima, and destroyed its usefulness to the enemy. That bomb has more power than 20,000 tons of T.N.T.
The Japanese began the war from the air at Pearl Harbor. They have been repaid many fold. And the end is not yet. With this bomb we have now added a new and revolutionary increase in destruction to supplement the growing power of our armed forces. In their present form these bombs are now in production and even more powerful forms are in development.
It is an atomic bomb. It is a harnessing of the basic power of the universe. The force from which the sun draws its power has been loosed against those who brought war to the Far East. …
We are now prepared to destroy more rapidly and completely every productive enterprise the Japanese have in any city. We shall destroy their docks, their factories, and their communications. Let there be no mistake; we shall completely destroy Japan's power to make war.
It was to spare the Japanese people from utter destruction that the ultimatum of July 26 was issued at Potsdam. Their leaders promptly rejected that ultimatum. If they do not now accept our terms they may expect a rain of ruin from the air, the like of which has never been seen on this earth. Behind this air attack will follow sea and land forces in such numbers and power as they have not yet seen and with the fighting skill of which they are already well aware. …
We have spent more than two billion dollars on the greatest scientific gamble in history-and we have won.
But the greatest marvel is not the size of the enterprise, its secrecy, nor its cost, but the achievement of scientific brains in making it work. And hardly less marvelous has been the capacity of industry to design, and of labor to operate, the machines and methods to do things never done before. Both science and industry worked together under the direction of the United States Army, which achieved a unique success in an amazingly short time. It is doubtful if such another combination could be got together in the world. What has been done is the greatest achievement of organized science in history.
For more information about the decision to use the atomic bomb, please see the resources on the Truman Library's website: https://www.trumanlibrary.gov/library/online-collections/decision-to-drop-atomic-bomb