Bath chief executive Bob Calleja said: "In a comparatively short time, Steve has made a very positive impression on everyone within the club.
"The decision to have a trial period was a great opportunity for Steve, a relatively young coach, to take on the responsibilities associated with the head coach role and see what the job involved.
"We are very pleased with the way that he has settled in. He has worked extremely hard, and has won the full support of the management and players."
Meehan added: "I am very happy to take on the full responsibility of this role, and that the club has seen fit to give me this opportunity fills me with confidence.
"I am looking forward to getting on with the job of helping the players at Bath develop to their full potential, which will lead towards the team becoming one of the top teams in England."
Bath are currently ninth in the Guinness Premiership with three wins from nine games, but are on course to reach this season's European Challenge Cup quarter-finals.