The 43-year-old was expected to be unveiled as Simon Grayson's successor following talks with the Championship side this week.
Chairman Dean Hoyle told the club's official website: "We've been through a lengthy and detailed process in appointing Mark as the club's new manager and we are delighted.
"As I said we would, we have taken our time in making this appointment because it is such an important one, both for our short and long term ambitions. Obviously retention our Championship status is vital, but we also needed a manager who buys into the club's football philosophy and structure. We believe Mark is that man.
"Mark has the knowledge, skills and attributes that the board of directors set out for our new manager and it is clear that he shares our vision, philosophy and values."
Robins's first game in charge will be Sunday's FA Cup fifth-round tie at home to Premier League outfit Wigan Athletic.