He is due to join McAlpine later in the summer, and takes over from Brian Melling, who is retiring.
A replacement for Smith is due to be announced next week.
Smith, who was at Mowlem for more than 20 years, said: "I am sad to leave but pleased to be going to a very good business. It has a very good civils business – it is well run and the market leader in terms of profitability and performance. The management wants someone to carry on running it well and make it even better."
A Mowlem spokesman said: "We wish Steve all the best in his new role."
Smith's appointment follows an upbeat trading statement by McAlpine on Monday. The firm said that prospects for the second half of the financial year, to 31 July, were encouraging and order books were strong.