I used the same trick with the previous GPT3 API (da-vinci) and it worked well, I'd pass as one big prompt:

  User: hello (previous prompt)
  Bot: hi (previous response)
  User: who are you? (new prompt)
  Bot: (here it continues conversation)

I wonder how the new ChatGPT API differs, other than the fact that it's structured (you use JSON to represent the conversation memory separately instead of one large prompt).

I guess I will spend the next day playing around with the new API to figure it out.