The 18-year-old youth-team graduate made his Stanley debut against Morecambe in this season's Johnstone's Paint Trophy and has committed his future to the club until the summer of 2015.
"I'm very pleased to sign to be honest," said Dawber in a club statement. "I was excited as soon as the boss (Leam Richardson) mentioned it to me so I'm just glad that it's gone through now and I can look forward to the football.
"It's a real vote of confidence to be given this contract and I'd like to thank the gaffer and everyone on the staff who has contributed towards it.
"I'm really happy to be staying here, I really like all the lads and I'm really pleased that everything is done.
"It's quite important to know that I can purely concentrate on the football and not have to worry about new deals in the summer.
"There are lads out there who don't have a contract and don't know their future so obviously I'm very happy to have that bit of security."