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.

The lawyer defending one of the robbers said they sometimes only saw half an hour of sunlight a day.
He was making submissions on the second day of a three-day sentencing hearing at Woolwich Crown Court.
But the prosecutor told the court their actions were likely to have caused "harassment, alarm or distress".
Since their arrests in May last year, they have been on remand in the high-security Belmarsh prison.
The gang - including men who were involved in the Security Express and Brinks Mat robberies - breached the vault at Hatton Garden Safe Deposit in central London over the Easter weekend last year.
Nick Corsellis, defending Carl Wood, 59, asked the judge to take into account the fact that Wood's inevitable jail term will mean that he will "not see" his ailing parents again.
He said his remand in custody had been "particularly difficult" because he and the other men are in the double A security category.
Mr Corsellis said it meant they are given one hour of exercise a day and "it is very rare that that is achieved - half an hour at best. Half an hour of daylight".
He claimed this had psychologically affected his client, saying: "The remand in custody has been difficult to bear."
The court also heard that Hugh Doyle, 49, had made an effort to turn away from crime, and that William Lincoln, 60, had a limited role in the conspiracy.
His lawyer, Dante Leccacorvi, said Lincoln "had a very distinct role and that was to conceal the proceeds of this crime on a short-term basis".
All the defence barristers argued that their clients should not be made the subject of Criminal Behaviour Orders.
Arguing for these to be imposed, prosecutor Philip Evans QC described the heist as a "significant burglary".
Adding that it was possible the men had also melted down the goods, or even taken them abroad, he continued: "These are sustained actions, and it is overwhelmingly likely that the victims of this offence would have been caused harassment, alarm or distress."
The men who pleaded guilty to conspiracy to commit burglary last September are:
Wood, of Elderbeck Close, Cheshunt, Hertfordshire, and Lincoln, of Winkley Street, Bethnal Green, east London, were convicted of conspiracy to commit burglary and conspiracy to conceal, convert or transfer criminal property after a trial.
Doyle, of Riverside Gardens, Enfield, was found guilty of concealing, converting or transferring criminal property between January 1 and May 19 last year.
Reader has been absent from the sentencing hearing after falling and suffering a stroke in Belmarsh.
All six men are due to be sentenced on Wednesday, when Judge Kinch will also decide when Reader will be sentenced.
Cabinet minister Greg Clark was given the wrong job by mistake during the appointment of Theresa May's new government.
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The men convicted of the Hatton Garden heist are facing tough conditions in prison, a court has heard.