The 17-year-old midfielder, who has tasted first-team football under manager Darren Ferguson this season, has put pen to paper on an 18-month deal at London Road.
He told the club's official website: "I am pleased to sign my first professional contract with Peterborough United. It is a great feeling.
"I have really enjoyed my time with the club so far and am very much looking forward to the future. I am fully aware that I have a long way to go and I need to continue to work hard on the training pitch."
Conlon, who was signed from Newcastle Town, is one of six players to have come through the youth ranks at Peterborough to feature in the first team this season.
Kgosi Ntlhe, Joe Newell, Jermaine Anderson and Jaanai Gordon have all appeared in the starting line-up, while teenage centre-back Liam Marshall and Irish midfielder Robert Duggan have featured on the substitutes bench.