The Championship strugglers had agreed the deal earlier in the week and have now finalised the paperwork to bring in the 20-year-old.
Livermore spent the first half of the season on loan at Derby County, scoring once in 16 appearances for Nigel Clough's side.
He will now join Posh for a training camp in Tenerife and could make his debut for the club against Derby next weekend.
Posh boss Mark Cooper told the club's official website: "Jake will give us a bit more bite in midfield, which will certainly give us another option.
"He is a box-to-box midfielder and we are delighted to have him on board."