Mr Clark heads up the new Department for Business, Energy and Industrial Strategy.
He was also appointed President of the Board of Trade before officials realised International Trade Secretary, Liam Fox, should hold the title.
Mr Clark lasted just four days in the historic role.
Once officials realised the mistake, Mr Fox was formally appointed to the position at a meeting of the Privy Council.
A Whitehall source said: "The appointment was an error. The Privy Council meeting was held very quickly after the reshuffle and the final list of ministerial responsibilities had not been produced. The appointment was therefore corrected at the first opportunity."
The mistaken appointment was made days three days before Mrs May told Parliament in a written statement that Mr Fox's new department would "take on the responsibilities of UK Trade and Investment (and) the relevant trade functions of the former Department for Business Innovation and Skills".
The Board of Trade - which is still formally named "The Committee of Privy Council for Trade and Foreign Plantations" - was established in the 17th Century by the King as a rival to Parliament to manage British trade, particularly in the American colonial plantations.
William Pitt created the job of president in 1786. After the board was folded into the modern-day Department of Trade and Industry in 1970, the historic board has met only once, in 1986, to mark its 200th anniversary.
Lord Heseltine chose to be known as President of the Board of Trade when he was appointed to John Major's cabinet in 1992, instead of Secretary of State for Trade and Industry.
Cabinet minister Greg Clark was given the wrong job by mistake during the appointment of Theresa May's new government.