Still, constitutions can be amended, if not ignored. In the case of Argentina, a two-thirds majority in both houses of Congress would be enough to allow 63-year-old Mr Menem to run again.
The trouble is, his ruling Peronist party has nowhere near such a majority. Results of yesterday's mid-term legislative elections, for half the seats in the lower house, were expected to create little change in the chamber, where Peronists hold 119 of the 257 seats.
The ballot, and any small swing either way, was nevertheless being watched as a barometer both of Mr Menem's popularity four years after his election as President and of his chances of winning moral support for his bid for a further term. Defying widespread opposition to the idea even within his own party, he has proposed a referendum, possibly as early as next month, on the constitutional amendment he needs.
A plebiscite, in which the President claims he would win 70 per cent support, would not be binding and would in itself change nothing. But it would most certainly stoke Mr Menem's obsession with staying on. 'Obsession' has become a word so overused by media interviewers that Mr Menem has taken to denying it before it is mentioned. 'He'll be coming out in an 'Honest, I'm Not Obsessed With Staying On' T-shirt soon,' ventured a Western diplomat in Buenos Aires.
Opposition leaders say he wants to become 'Emperor Carlos I' or 'President-for-life,' and mock him in a television advertisement showing him posing alongside a Ferrari, a small gift from an admirer. Others call him el Moro (the Moor) in reference to his Arab origins.
'Four more years,' Mr Menem responds, echoing US party cheerleaders. He refers to four years beyond 1995, when his current term expires. 'I need 10 years to complete my project,' he says, noting that Francois Mitterrand and Felipe Gonzalez are already in double figures.
The success of his economic project, rather than his personality and despite his trustworthiness- rating with the public, leads many Argentinians to believe he might just pull it off and lead the country into the 21st century. When he took over from Raul Alfonsin in 1989, the country was virtually bankrupt and inflation was at an annual 20,000 per cent.
He has slashed that to a level many European countries would envy - consumer prices did not rise at all from July to August this year - and stabilised the peso by pegging it to the dollar. The poor in the villas (shanties) around Buenos Aires, who were forced to ransack supermarkets towards the end of Mr Alfonsin's term, now at least have the impression they are not getting poorer by the day, though Mr Menem has failed to improve health care or education.
The well-off, meanwhile, are even better off. They can holiday in Brazil or fly to Miami for a change of wardrobe without changing trillions of pesos or dipping too deeply into their US bank accounts.
Whether the economic improvements are enough to overcome a widespread image of corruption remains to be seen.
A weekend scandal involving the nine-man Supreme Court - six of them Menem appointees - won more headlines than the mid-term elections.
Two of the non-Menem justices said a court ruling unfavourable to the Central Bank had been stolen and replaced with one amended in the bank's favour.
Mr Menem's right-hand man, the Economy Minister, Domingo Cavallo, responded by suggesting that one of the two justices, Augusto Belluscio, may have killed his lover in Paris in 1989.
'It just so happens he (Belluscio) was in his underpants in the room with the woman who jumped out the window,' the daily Clarin newspaper quoted Mr Cavallo as saying.