The 17-year-old has been a key player for their Under-18's this campaign and the Dons were keen to reward his efforts with a permament deal.
Bamba told the club's official website: "I am very happy to get this deal. It is something I have dreamed about since I was very young.
"To have signed my first contract is very exciting and promising. I am just grateful that the club have enough trust in me to give me a contract."
Manager Neal Ardley said: "Youssef came in on my first day and he was looking for a trial. (Youth coach) Mark Robinson has done really well with him and gave him a chance.
"Since then he has got fitter and stronger and I watched him score two goals against Cheltenham Town in the FA Youth Cup. He put on a very good performance that night and Mark has been very impressed by him.
"He was not on a contract and we just felt it was a no-brainer to sign him up and see how he develops."