Stone, who left his position as commercial director at Laing O'Rourke abruptly in June 2003, started in the role last Thursday. In the past year he has acted as interim commercial director for Birse as well as offering consultancy for firms such as Sir Robert McAlpine, Bluestone and QS Ridge.
Stone, who originally joined Laing from school 30 years ago, said the Gleeson job had come about by "pure chance".
He said: "I wasn't even looking for a new job. One of the Birse people joined Gleeson and when Andrew [Muncey, Gleeson group managing director] told him he wanted a commercial director he mentioned my name.
"I met Andrew and got on extremely well with him. The fact that Gleeson has never had anyone in this position makes it quite a challenge."
Stone said he would look after risk management, estimating, procurement, internal auditing and surveying at the £620m-turnover company, which has separate contracting, housing and property operations.
There was never a question of me clearing my desk
He said: "I think the firm is looking for someone to bring consistency across the businesses and raise the standards in some parts.
"The business has grown rapidly in the past five years but it has grown in individual ways rather than in a co-ordinated fashion. We are also trying to make the business better at forecasting and reporting what it does."
Stone said he would oversee the housing and property arms but his primary focus was construction.
Stone described his departure from Laing O'Rourke as "very amicable" and denied reports he was given an hour to clear his desk when he left.