Small won his deal after successful trials.
Bournemouth academy chief Alan Connell told the club's website: “We are really pleased to bring in Coby. He trained with us for a few weeks during pre-season and impressed, especially in a friendly against Southampton.
“He’s a really focused young man who wants to improve his game and we think he’s got the potential to do that with us.
“Now, it’s up to myself, the staff and, most importantly, Coby to maximise his potential and we are all excited to see how he gets on.
The 18-year-old said: “I am buzzing to be signing here. After coming down and seeing all the boys and playing for them, I just think it’s the perfect fit for me.
“Playing here against Burnley was a tough game but a very good experience and to win it in the last minute is always nice.”