But what are the consequences of this shock first round result? What will it mean for Slovakia's image abroad? Will it have implications for Mikulas Dzurinda's already shaky ruling coalition? Radio Prague's Rob Cameron has been speaking to analyst Ivo Samson, from the Bratislava think-tank the Slovak Foreign Policy Association. He began by asking him to explain why Mr Meciar had done so well, and why Mr Kukan had done so badly.
"The governing coalition was not able to concentrate on one candidate. Second, all the opinion polls showed that Mr Kukan would either win the first round or come second, and that he would go into the second round against a representative of the opposition, probably Vladimir Meciar. And a lot of people believed it wasn't worth going to the ballot box in the first round and would wait until the second round."
Prime Minister Mikulas Dzurinda has said he will be pondering not only the future of Slovakia but also his own future in the wake of this defeat. Will his government survive this fiasco do you think?
"There is a lot of discussion about this. I believe his government has the chance to survive under the condition Mr Dzurinda evolves personal consequences and resigns. Most people in his Slovak Democratic and Christian Union might point out that it was the behaviour of the prime minister which is to blame for this humiliating defeat."
So Mr Dzurinda's days could be numbered.
"If he loses ground in his own party, then it might be true."
Slovakia joins the European Union in less than a month, and has just joined NATO. Yet the country's next president will either be Vladimir Meciar or Ivan Gasparovic, two men who arguably did much to sabotage Slovakia's membership of those two organisations. It's an extremely ironic situation isn't it?
"It is. There is of course a slight difference between Mr Meciar and Mr Gasparovic. Mr Gasparovic is as good as unknown abroad, so he cannot be blamed for the bad image of Slovakia before. Mr Meciar would be a bigger shock for public opinion and for experts abroad. But if you go into details, Mr Gasparovic - the former chairman of parliament under Mr Meciar - could be blamed for the development in Slovakia between 1994 and 1998, including the abduction of the son of the former president. He backed it at least."
Surely that's an unacceptable situation for many Slovaks?
"I believe Slovaks are shocked now. They know it is an unacceptable situation but unfortunately they lost the chance to change it because they didn't go and vote."
Do you think the rest of Europe, the international community, NATO and the European Union can work with Slovakia with Mr Meciar or Mr Gasparovic as president?
"Well it's not the prime minister, it's not an important figure in the state. It's a symbolic head of state, and his powers are very limited. So I believe EU countries and NATO countries, and NATO and the EU as a whole, can co-operate with Slovakia. But it will be very unpleasant for both the EU and NATO."
And even they must be rather shocked.
"They surely were shocked. And still are in shock."
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