The episode demonstrated a number of things. First, that even in extremis the President conserved his acute sense of what appeals to his people. And second that there are many aspects of the flamboyant President's personality that are often kept from the public eye. But Mr Menem himself has always taken the view that the more extravagant his personal excesses, his amorous adventures, his love for fast cars, his desire to 'get down on to the pitch', the more his people will love him.
Since he was elected president in 1989 he has taken the view that his was a long-term project. He needed time, he said, for his economic reforms to bear fruit and he began early to prepare the way for staying in power beyond the statutory six-year term which ends in 1995.
Sunday marks an important milestone in the achievement of this ambition. Argentina is to elect 350 members of a constituent assembly whose duty will be to reform the country's 1853 constitution. Its key task will be to change the constitutional requirement that the presidency be limited to a single six-year term. It will be asked to approve the re- election of a president for a second term, and to restrict the presidency to four years.
Other proposed reforms are to create the post of Prime Minister, which would pull some power in Argentina's executive-heavy political system back towards parliament, and would also lighten the workload of the President.
It is not the first time Argentina's constitution has been tinkered with. Mr Menem's predecessor, Juan Domingo Peron, changed it in 1949 to enable his re- election in 1952, but his reforms were swept away by the generals who succeeded him.
Mr Menem's bold free-market economic policies seem the obverse of the state welfarism of his party's founder, and many of Peronism's natural supporters would seem to be alienated by the ruthless stripping away of the huge state sector and the welfare state. But he continues to be popular, and has won new supporters from unlikely quarters. Many among Argentina's traditional elite who were snobbish about his immigrant origins - his father was a Syrian immigrant - and his vulgar style now admire him for achieving economic stability, unknown to Argentines for a generation.
His appeal owes more to the vulgar glamour of Evita than the heavyweight charisma of Juan Domingo - although Mr Menem is a dynamic public speaker - but his support hangs on the continuation of the success of his economic programme.
'I have always loathed Peronism,' said Manuel de Anchorena, from one of the country's oldest families, a man who served the military dictatorship as ambassador to France. 'But I have to hand it out to Menem. He has achieved things that we during the military regime would not have dared to attempt.'
Mr Menem is, for the moment, Peronism's sole candidate; he has no convincing rival within his party. The constitutional reforms, therefore, which must be concluded by August, will probably sail through. Main points of contention with the opposition Radical Party were dealt with last November during Mr Menem's 'Pact of Olivos' with the Radicals' leader, Raul Alfonsin. The main outcome of this deal was an agreement to strengthen the Radicals' presence among judges in the Supreme Court, going some way to meeting criticism that the judiciary was over-dependent on the Peronist party.
Mr Menem must be hoping a second presidency does not resemble that of his predecessor. Juan Peron's second term was disappointing: beset by economic crisis, his support dwindled and he was kicked out by the generals.
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