The leadership of the government ministries was assigned as part of the coalition agreement between Mr. Kurz’s conservative Austrian People’s Party and the far-right Freedom Party.
Any changes require the approval of the country’s president, Alexander Van der Bellen, who called on Sunday for new elections in September. He will now need to appoint technocrats in charge of the abandoned ministries until a new government takes power.
“We will fill the free positions with experts or high-ranking civil servants to ensure that the government remains able to govern until the election,” the chancellor said.
The scandal of recent days has brought an abrupt end to the government that Mr. Kurz, 32, had formed with the Freedom Party amid criticism that he was enabling the nationalist, far-right party after only 17 months in power.
The Freedom Party was founded by former Nazis in the years after World War II, and has long pushed a nationalist, anti-immigrant agenda that Mr. Kurz co-opted, repacked and rode to victory in elections in 2017. He has since depended on their support.
The Freedom Party’s ties to far-right extremists, including Generation Identity, a group that is under investigation by several European intelligence services, are well documented.
The video showing Mr. Strache and another member of his party was filmed without his knowledge. The footage was obtained and published on Friday by the German newsmagazine Der Spiegel and the newspaper Süddeutsche Zeitung, two respected news outlets.