Mowbray said: "The ink on the contract is drying as we speak.
"I had a conversation with Stephen this morning and he is looking forward to the challenge.
"We wish him well, it's a club I know well and I hope he does well for them, I'm sure he will."
The Celtic boss admitted he has had talks with Scott McDonald who has also been linked with a move to Middlesbrough and West Ham.
He said: "McDonald has got to get on with his rehabilitation.
"I won't sit here and deny there has been conversations with Scott and they have been interesting chats but at the moment there is nothing to say either way.
"He is our top scorer and I would prefer to keep him."