Video from an interview in 1965.
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".Further explanation at Factoids, via Could it be Madness - this?
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.Bhagavad Gita, chapter 11, verses 31-33
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.