Greece names first woman as interim PM.
President Prokopis Pavlopoulos named Ms Thanou after efforts to form a coalition failed. Last week, Alex Tsipras resigned as prime minister to seek a new mandate for office.
Ms Thanou, 65, becomes Greece's first female prime minister.
Elections are expected to be scheduled for 20 or 27 September.
Ms Thanou's appointment ends a week of fruitless negotiations as opposition party leaders tried unsuccessfully to form a government.
Appointed president of the Supreme Court in June, days before Greece'sYes/No referendum. She is the second woman to have held the role
Sent a letter to Jean-Claude Juncker, head of the European Commission, in February urging support for the new Syriza government's reform plans
Wrote an open letter during bailout talks in July expressing her belief that most Greeks wanted to remain in the eurozone
Criticised the leadership of then Prime Minister, Antonis Samaras, in October 2014, condemning his "totalitarian government"
Mr Tsipras stepped down as prime minister and called early elections after 25 of his MPs quit Syriza over the bailout he agreed with European creditors and formed the left-wing Popular Unity party.
In a statement live on television last Thursday, he said it was now up to the Greek people to give their verdict on whether to continue with his government's programme.
Mr Tsipras is expected to win the next election although it is unclear whether he will secure a majority government.
However, he has ruled out a coalition with any of the more centrist opposition parties: centre-right New Democracy, the socialist Pasok party or the small centrist The River (To Potami) party.
Earlier this week an opinion poll for Greece's Vergina TV suggested support for Mr Tsipras's Syriza party had declined to 24%, down from 34% in July.
New Democracy was in second with 22%, while the far-right Golden Dawn ranked third with 6%.
Popular Unity, which split from Syriza, was put on 4.5%.
Panagiotis Lafazanis, who formed Popular Unity, was the last of three party leaders who were given the chance form a government in the past week.
He used the opportunity to air his anti-bailout message before handing back the mandate to the president on Thursday.
Varoufakis 'lost credibility'
Former Finance Minister Yanis Varoufakis decided not to join Popular Unity despite his opposition to the €86bn (£62bn; $96bn) eurozone bailout agreed by his successor Euclid Tsakalotos.
Speaking to Australia's ABC News on Wednesday, Mr Varoufakis said he would not be running for the Greek parliament in the September elections, as he no longer believed in what Syriza and Mr Tsipras were doing.
For his part, Mr Tsipras hit out at his former finance minister late on Wednesday, telling Alpha TV that he had realised in June that "Varoufakis was talking, but nobody paid any attention to him".
Negotiators in the crucial debt talks with the IMF and European Union lenders "had switched off, they didn't listen to what he was saying", he said.
"He didn't say anything bad but he had lost his credibility among his interlocutors."