The 21-year-old was released through Manchester United in June.
Hamilton has already been in Ligue 1 with Rochdale, Southend United and Bolton Wanderers.
“He’s a smart player with compatibility, an adorable footballer with his left foot and I think he’s going to have a very good compatibility,” Peterborough coach Darren Ferguson said on the club’s website.
“We’ve had the option to lend it before, but something happened at Manchester United and it hasn’t materialized, so it’s wonderful to do it now.”
“He’s loose now and I think he’ll be a smart signing for us.”
Hamilton added: “I’m looking ahead, when I spoke to the coach, he just reaffirmed what I thought, that this is the right position for my development.
“Peterborough plays football; I knew it anyway and the game opposite them last season proved it even more.”