Final answer:
The correct conjunction to use in the sentence is 'that'. It signifies the consequence of being extremely tired, which in this case is falling asleep immediately.
Step-by-step explanation:
In English grammar, the correct conjunction to use in the sentence 'She was so tired _______ she at once fell asleep' is 'that'. The conjunction 'that' is used in this case to indicate the result of something. Here, the result of her extreme tiredness caused her to fall asleep immediately. Therefore, the completed sentence should read: 'She was so tired that she at once fell asleep.'
Learn more about English Conjunctions