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Three, one that was tested in the US, one was deployed at Hiroshima and the third at Nagasaki. Bombs were under construction, but no further weapons were ready for use by the surrender date. A fourth bomb was made by Los Alamos before Japan agreed to surrender, but did not arrive in San Francisco to be flown across the Pacific until August 18, four days after the Japanese agreed to surrender (but before the formal signing of the agreement which happened in September). This is confirmed by declassified documents obtained by Chuck Hansen via FOIA requests and summarized in his work Swords of Armageddon.

As 3 more bombs were scheduled for production and delivery in September (ibid.) it is almost certain that Los Alamos completed the first of these 3 within a few days before or after the formal signing. However documents on the completion date of this bomb have not been released. If they did complete this bomb prior to the signing, this would make five in total were built during WW2.

The next definite data in Swords of Armageddon gives bomb production up to the end of the 1946 Operation Crossroads: total bombs built 9, total bombs detonated 5, bombs remaining in stockpile 4.

Note: bombs of this era when built were not assembled. for example a MK-III Fatman came in two crates of parts and took 3 days of final assembly in the field before use.

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Q: How many atomic bombs were made during ww2?
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