Alex Salmond revealed that former Lord Advocate and senior judge Lord Hardie is to lead the inquiry.
He said Lord Hardie would start work soon and added: "We look forward to a swift and thorough inquiry."
The First Minister announced to Holyrood last week that there would be a judge-led inquiry into the controversial project.
Trams began running in Edinburgh at the end of last month after years of delays, spiralling costs and a lengthy dispute between the city council and its contractor.
The scheme to return trams to the streets of the capital, with construction of a line from Edinburgh Airport to York Place, has cost around £776 million.
The inquiry will examine the project's governance, management and other areas to try to find out why it was delayed, and why it "cost considerably more than originally budgeted for and delivered significantly less than was projected through reductions in scope".
It will also consider the consequences of the failure to deliver the project on time and on budget, as well as making recommendations on how similar projects could avoid such problems in the future.
Lord Hardie will appoint a team to carry out the inquiry and set a timetable for the work.
Meanwhile, dozens of day-tripping pensioners are tucking into lunch at Edinburgh Airport each day as free tram travel for the over-65s begins to take off.
Scotland's busiest terminal has seen an unexpected boon in lunches since the tram launch on May 31.
Food and drink sales at the airport have soared by 8% in recent weeks, while tram ticket collectors have told of a daily exodus to the airport by retired residents.
John Watson, the airport's chief commercial officer, said there had been a "significant increase" in families and pensioners visiting "simply to experience the tram for themselves, without flying".