The 18-year-old goalkeeper has been granted a one-year deal, plus a further year's option in the club's favour.
Sporting & Technical Director (Designate) Richard Garlick told Albion's website: "Jack is one of a number of good goalkeepers at the club. We are well off in that department.
"Jack will get good opportunities in our Under-21s team. This is an important year for him."