Alexander Van der Bellen, who beat the far-right candidate in a presidential election thriller last year, was sworn in on Thursday just as speculation raged about the future of the government.
Van der Bellen, 73, narrowly won a run-off against Norbert Hofer from the Freedom Party (FPOe) in May, but the anti-immigration party got the result annulled due to procedural irregularities.
A re-run set for October had to be postponed because of faulty glue on postal votes and after an ugly and divisive campaign Van der Bellen finally won the largely ceremonial, but coveted presidency on December 4.
Hofer, 45, who like similar politicians in Europe - and Donald Trump in the United States - stoked concerns about immigration and security, would have been the first far-right president in the European Union.
The victory of the professorial Van der Bellen, who was backed by the ecologist Greens but ran as an independent, was greeted with relief by centrist parties in Austria and around Europe.