On paper, the picture is clear. The Russian constitution, adopted in December 1993, states that in the event of the President's death, or incapacity to fulfil his duties, his job passes temporarily to the Prime Minister, currently Viktor Chernomyrdin. The Prime Minister is then obliged to call fresh presidential elections within three months.
In practice, there is no certainty that Russia would easily surmount the upheaval provoked by Mr Yeltsin's premature departure from office. The constitutional mechanisms that are in place have never been tested and may count for little against a centuries-old tradition of power struggles, often violent, that have accompanied the demise of a tsar or party chief.
Moreover, whereas the constitution is considered almost sacred by the political classes in a country such as the United States, there is no such devotion in Russia to a document that is widely seen as having been tailor-made for Mr Yeltsin. He drew up the constitution in the aftermath of the armed uprising in the Russian parliament building in October 1993. The extraordinary range of powers that it granted him at the expense of the legislature was designed to ensure that no one could mount a serious threat to his rule again.
The fragility of Russia's constitutional order was exposed shortly after the first round of the presidential election, on 16 June, when a cabal of hawks including the Defence Minister, the head of the former KGB and Mr Yeltsin's personal security chief were drummed out of the Kremlin on charges of trying to force the postponement of the election. Less than a month before that drama, Mr Yeltsin had raised doubts about his willingness to abide by constitutional procedures when he rejected a law passed by the Communist-dominated parliament that set out the process by which he would hand over power should he lose the election.
Many Russian political commentators believe that, despite insisting that the election should take place, Mr Yeltsin never intended to make a graceful exit from office if defeat loomed as a realistic prospect. Moscow has buzzed with rumours of a so-called "Plan B", according to which Mr Yeltsin would have declared a national emergency and stayed in power rather than vacate the Kremlin for Gennady Zyuganov, his Communist challenger.
Inevitably, the impression that the constitution is just a piece of paper to be altered, ignored or scrapped at will has fed through into the attitudes of many prominent Russian politicians. For no one is this more true than for Alexander Lebed, the outspoken former general whom Mr Yeltsin put in charge of national security, after Mr Lebed finished third in the election's first round.
Despite his lack of a genuine power base in Mr Yeltsin's obscure Kremlin power structures, Mr Lebed has not disguised his ambition to rule Russia as soon as possible.
He proposed last week that he should be given the post of vice-president, a job which was abolished in 1993 after its then incumbent, Alexander Rutskoi, took part in the armed revolt at the White House. Mr Lebed clearly sees himself as the heir-apparent, constitution or no constitution.
Other influential figures in Mr Yeltsin's entourage are likely to take a different view, especially since Mr Lebed has revealed in the past two weeks that his opinions are much more illiberal than he indicated during his election campaign.
Those who might resist a Lebed bid for power include not only reformists, such as Anatoly Chubais, one of Mr Yeltsin's top campaign strategists, but more centrist politicians with a taste for power and patronage, such as Mr Chernomyrdin.
Last July, when President Yeltsin suffered the first of his two heart attacks in 1995, Mr Lebed was a marginal political figure and Mr Chernomyrdin was the only plausible president-in-waiting. Now Mr Lebed holds centre stage.
If Mr Yeltsin's health continues to decline it is difficult to see how Russia can avoid yet another of its periodic clashes for power in the Kremlin.