Weir, 23, who was released by Tranmere at the end of last season, becomes manager Gary Rowett's first permanent signing since he was handed the job on a permanent basis earlier this month.
"I am delighted to be able to bring Robbie to Burton Albion," said Rowett. "Robbie is a real box-to-box midfielder and full of energy. He works hard for the team and has that athleticism that I think we need. We have been missing someone like Robbie probably since Russell Penn left last summer, that is the sort of player he is.
"I think he was a little unlucky to be released from Tranmere. He played 40-odd games for them last season in League One so has the experience at a higher level that we are looking to try and bring in.
"There has been a bit of interest in him from League One clubs so we are pleased to have got him here. He is a good age as well and I am sure he is going to be a success for Burton Albion."