Colleagues of the British minister involved set up a diversionary cover story to explain his absence, saying he was off to Switzerland to do a little painting with his wife.
The secret meeting is disclosed in the official history of the Falklands by Sir Lawrence Freedman, professor of war studies at King's College, University of London.
He reveals that in June 1980, the Foreign Office drew up a proposal, approved by the cabinet's defence committee, whereby Britain would hand Argentina titular sovereignty over the islands, which would then be leased back by Britain for 99 years.
The British and Argentinian flags would be flown side by side on public buildings on the islands. British administration would continue with a view to guaranteeing the islanders and their descendants "uninterrupted enjoyment of their way of life".
The driving force behind the plan was the Foreign Office minister Nicholas Ridley.
He proposed a secret meeting with his Argentinian opposite number, Comodoro Cavandoli, in Venice in September 1980. He would be accompanied by his wife, ostensibly on a private holiday.
However, Lord Carrington, the foreign secretary, was worried about the choice of venue. "Why Venice?" he asked. "It all looks very hole in the corner."
Eventually, a Swiss location was chosen - the Hotel du Lac, situated in the picturesque lakeside village of Coppet, about 10 miles from Geneva.
The government invented a cover story - that "Mr Ridley's visit to Geneva with his wife is private, for a short holiday break, and that he hopes to do a little watercolour painting".
Mr Ridley had already agreed the sale of Lynx helicopters and naval missiles to Argentina and he and Mr Cavandoli seemed to enjoy a mutually warm relationship. Their meetings in Switzerland appeared to go well - certainly, Mr Ridley thought so - and they met again in New York soon afterwards.
However, the plan was wrecked after Mr Ridley, whose mission was not helped by a rather offhand and patronising manner, made an ill-fated trip to the Falklands in November, where he tried to sell a deal to the islanders. Suspicion about the government's long-term intentions grew, fuelling opposition among both Conservative and Labour MPs to any such deal.
Sir Lawrence also reveals how the Thatcher government came under unrelenting pressure from Washington to agree a ceasefire after the Argentinian invasion and before the islands had been recaptured.
Lady Thatcher and President Ronald Reagan were engaged in heated exchanges as Washington's concern about its relations with South American countries led to strong pressure on Britain to come to a deal with the Argentinian junta.
Alexander Haig, the US secretary of state, proposed a ceasefire with an international peacekeeping force, including US troops. Lady Thatcher told Reagan in a telephone call at the end of May 1982 that Britain could not contemplate a ceasefire before Argentina withdrew from the Falklands.
According to Sir Lawrence, she asked Reagan: "How would the Americans react if Alaska were invaded and, as the invaders were being thrown out, there were calls for the Americans to withdraw?" She is said to have been "dismayed" by Reagan's attitude and wanted him to know just how "upset" she was.
Washington pointed out that the US had secretly supplied Britain's special forces with communications satellites and ammunition. But Lady Thatcher was adamant. "We have lost a lot of blood, and it's the best blood," she told Sir Nicholas Henderson, Britain's ambassador to the US, on an open line. "Do they not realise," she added, "that it is an issue of principle? We cannot surrender principles for expediency."
Meanwhile, France from the start proved to be Britain's staunchest ally.
But today's official history makes clear that Britain's claim to the Falklands is not as strong as has been made out. Sir Lawrence points out that Britain relied, not on prior discovery, but on a small settlement established in 1766 but abandoned in 1774. When Britain recognised Argentinian independence in 1825, it did so without any claim to the Falklands, which were then under an Argentinian governor living there.
Britain seized the islands by force in 1833.
· The Official History of the Falklands Campaign, Sir Lawrence Freedman, published by Routledge Taylor & Francis Group, volume 1 (£ 39.95), volume 2 (£49.95).
Islands under occupation
1833
Britain seizes control of the Falklands
1982
April 2 Argentina invades the Falklands
April 5 British taskforce sets sail as diplomatic talks are started
May 2 Belgrano sunk in the first major action of the war
May 21 British military forces land on the Falklands near Port San Carlos and head towards Darwin and Goose Green
June 14 British recapture Falklands - the union flag flies over Port Stanley
Taskforce 102 ships
British forces deployed in the conflict - 29,000
Casualties
British: 253 dead
Argentina: 750 dead
Medals awarded 151 including two VCs
Distance from Britain to Falklands 8,000 miles