And A Darkness Fell Over The Land - I Am Become Death

"I am become death, the destroyer of worlds"
J. Robert Oppenheimer, Trinity 1945
Robert Oppenheimer was the director of the laboratory in Los Alamos, New Mexico, and the scientific director of the Manhattan project. Since so many talents were involved it's somewhat misleading to call him "the father of the nuclear bomb", but he undeniably made one of the major individual contributions.
In an interview from 1965, Oppenheimer describes the initial reactions as the fruit of their labors, the very first nuclear bomb (the Hiroshima bomb was the second one), detonated early in the morning of July 16, 1945: 

We knew the world would not be the same. A few people laughed... A few people cried... Most people were silent. I remembered the line from the Hindu scripture the Bhagavad Gita; Vishnu is trying to persuade the prince that he should do his duty, and to impress him takes on his multi-armed form, and says, "Now I am become death, the destroyer of worlds." I suppose we all thought that, one way or another.
The quote was something he thought, but he didn't say it. 

The quote is indeed from the Bhagavad Gita ("Song of the lord"). Some suggest it's a misquote, which would explain the peculiar grammar; but "am become" is not an error but a (poetic) archaism, as in "I am become a name, for always roaming with a hungry heart" (Tennyson, Ulysses). Which in turn might be a trace of French; "Je suis devenu la mort".
Since Oppenheimer was proficient in sanskrit he read the original text, and the translation is his own; I haven't found any other translation with "am become". It certainly gives a certain something to the line, however, and it might had been at least somewhat less well known had it been "I am death" or "I have become death".
Here's the verse in question with a little context (translated by Ramanand Prasad). Prince Arjuna hesitates to attack the enemy with his army; Vishnu, in the incarnation of Krishna, encourages him, and motivates him by explaining how the world works, with reincarnations, Brahman, Maya etc. Arjuna asks to see Vishnu in his "cosmic", i.e. real, form, a wish that is granted. The overwhelmed Arjuna asks:

Tell me who are You in such a fierce form? My salutations to You, O best of gods, be merciful! I wish to understand You, the primal Being, because I do not know Your mission. The Supreme Lord said: I am death, the mighty destroyer of the world, out to destroy. Even without your participation all the warriors standing arrayed in the opposing armies shall cease to exist. Therefore, get up and attain glory. Conquer your enemies and enjoy a prosperous kingdom. All these (warriors) have already been destroyed by Me. You are only an instrument, O Arjuna.

Bhagavad Gita, chapter 11, verses 31-33

In an ancient Hindu scripture one might expect something a little less violent, but apparently the word that is here translated as "death" can also be interpreted as "time", which softens the message a little, at least if you're Hindu. The word is kala, which can mean "time" or "dark". The feminine form is Kali, the infamous goddess of death.

Sri Bhagavan uvaca / Kalo asmi loka-ksaya-krit
pravardho / lokan samartum iha pravattah ...

Here's another description of the event, where...

...two pages from the Bhagavad-Gita, the sacred Hindu poem, flashed through Oppenheimer's mind: "If the radiance of a thousand suns / were to burst into the sky / that would be like / the splendour of the Mighty One" and "I am become Death, the shatterer of worlds".

Current Biography Yearbook 1964

The first verse (rather than page...) mentioned is a part of the description of the cosmic form of Vishnu, and is found in verse 12 in the same chapter as "become Death". The citation from 1964 is the oldest I've found with it. The two verses are often erroneously combined into a single one.

If the splendour of a thousand suns were to blaze out at once (simultaneously) in the sky, that would be the splendour of that mighty Being (great soul).

Bhagavadgita 11:12 (Sivananda)

In one translation, there's a negation:

If the splendor of thousands of suns were to blaze forth all at once in the sky, even that would not resemble the splendor of that exalted being.

Bhagavadgita 11:12 (Prasad)

Since I haven't found any older descriptions with verse 12, and it isn't in Oppenheimer's own description, I draw the conclusion Oppenheimer didn't think of it when the bomb went off.

He grew tenser as the last seconds ticked off. He scarcely breathed. He held on to a post to steady himself ... When the announcer shouted 'Now!' and there came this tremendous burst of light, followed ... by the deep-growling roar of the explosion, his face relaxed into an expression of tremendous relief.

The reaction of Oppenheimer, as described by Brigadier General Thomas F. Farrell.

And I wish I would remember what my brother said, but I can't - but I think we just said, "It worked." I think that's what we said, both of us. "It worked".

Frank Oppenheimer was the brother of J. Robert, was involved too in the Manhattan project, and was on location in Trinity.
Note 1: In Full metal jacket, the soldier Animal Mother has "I am become death" written on his helmet.
Note 2: The novel A Thousand Splendid Suns by Khaled Hosseini does not derive it's title from the Bhagavad Gita, but from a poem about Kabul by Saib-e-Tabrizi, a Persian poet in the 17th century.

Sources:
Wikiquote: Interview with Robert Oppenheimer, from the The Decision to Drop the Bomb, 1965
The short clip from the interview can be found all over the net, like here or here
"Bomb Peril Cited by Oppenheimer", New York Times May 31 1955 - quotes an article from Le Figaro
Bhagavad Gita translated by Ramanand Prasad
Bhagavad Gita translated by Sri Swami Sivananda
Bhagavad Gita in original sanskrit [PDF]
Everything2.com: Now I am become death..., with 11:32 translated by Swami Tripurari
"The Eternal Apprentice", Time, November 8 1948
Current Biography Yearbook 1964, page 331
Richard Rhodes, The making of the atomic bomb (Penguin Books 1988), p 675 - quotes Frank O. 

SOURCE