The goalkeeper joined on a short-term agreement towards the start of this term and has been penning month-to-month deals having spent a lengthy period on the sidelines with a serious knee injury between October and January.
But Brewers boss Gary Rowett has seen enough of the fit-again 27-year-old, who has managed 14 appearances either side of his injury setback, to offer him an extended deal.
"It's terrific that we have now got Stuart signed up not only for the remainder of this season but also for next season as well which is important," Rowett told the club's official website.
"He did really well for us in the first part of his spell when we went eleven games unbeaten.
"He got his injury but he has come back really quickly from that and played the last three games where we have been unbeaten again, including two away games where our form of late hasn't been good.
"But hopefully that form is improving now and Stuart can have a big part to play again."