Postal ballots will decide Austria's presidential election after polling station results from Sunday's vote gave the far-right candidate a slender lead.
Norbert Hofer of the Freedom Party was slightly ahead of his rival, Alexander Van der Bellen, the interior ministry said on Sunday.
If elected, Mr Hofer would be the first far-right head of state in the EU.
A key campaign issue was Europe's migrant crisis, which has seen asylum-seeker numbers soar.
About 90,000 people claimed asylum in Austria last year, equivalent to about 1% of the Austrian population, and the Freedom Party ran an anti-immigration campaign.
Some 750,000 postal votes from roughly 12% of Austria's 6.4m voters are due to be counted on Monday.
Austria's election has revealed a profound split over which direction the nation should take.
Support for the Freedom Party has risen in recent years because deep frustration with the established parties of the centre left and centre right. Fears about the migrant crisis have boosted the far right still further.
The presidency is a mainly ceremonial post, but a victory for the Freedom Party candidate Norbert Hofer would give momentum to populist and far-right parties in other European countries.
For the first time since World War Two, both the main centrist parties were knocked out in the first round.
The presidents of the European Commission and the European Parliament, Jean-Claude Juncker and Martin Schulz, have both expressed concern over a Hofer victory.
Mr Hofer had 51.9% and Mr Van der Bellen — an independent candidate who is supported by the Green Party — had 48.1% after the final count of votes cast at polling stations, the interior ministry said.
The eventual victor would have "the job of uniting Austria", Mr Hofer said.
Mr Van der Bellen said he had championed the EU during his campaign.
Vying to lead Austria
Background: Aeronautical engineer
Politics: Far-right Freedom Party
Campaign soundbite: "To those in Austria who go to war for the Islamic State or rape women — I say to those people: 'This is not your home'"
Alexander Van der Bellen
Background: Economics professor
Politics: Former Green Party leader
Campaign soundbite: "I've experienced how Austria rose from the ruins of World War Two, caused by the madness of nationalism."
In the first round last month, Mr Hofer secured 35% of the votes, while Mr Van der Bellen polled 21%.
The result prompted Chancellor Werner Faymann to resign after he lost the support of his Social Democratic party colleagues.
The Social Democrats and the People's Party have governed Austria for decades, either alone or in coalition.
At the last general election in 2013, they together won just enough votes to govern in a "grand coalition".
Incumbent President Heinz Fischer, 77, could not run again after two terms in office.