Sweeney, 17, made 12 appearances for Bray last season and is a Republic of Ireland youth international.
Sweeney follows in the footsteps of compatriots Shane Long and Kevin Dole, who arrived Reading from St Patrick's and Cork respectively.
He said: "I'm absolutely delighted, it's a big dream of mine and the best moment of my career so far.
"There are a good few Irish connections here which helped to attract me to the club. There are plenty of players who have come from the League of Ireland and made the step up at this club, breaking into the first team, so it's something I want to aim for."
Director of Football Nick Hammond said, "At the age of 17, Pierce is a young player with great potential and one that we've identified and watched over a long period of time.
"He is entering an excellent environment, and I know he will work extremely hard to try and realise his ambitions.
"The recruitment of young talent is massively important for us going forward, and after doing well in men's football in the League of Ireland he will now come into our development group."