The 23-year-old left-back, who has not appeared in Mansfield's first two matches this season, has been drafted in as cover for Reece James, on loan from Manchester United, who is nursing a thigh strain.
Cumbrians manager Greg Abbott told the club's official website: "Reece isn't going to be available for Saturday and it's an area we needed to cover.
"Paul is a solid defender who knows the game and he's another who is at a stage where he just wants to be playing games.
"We'll get him in and take it from there."
Black is available for Saturday's game at Bradford and has been given clearance for the Capital One Cup second round tie against Leicester later this month.