Turkey's secular establishment hopes that the new government will usher in an era of stability and end the perceived threat from the country's Islamists.
Mr Demirel's decision comes as a grave disappointment to Tansu Ciller, the departing foreign minister. She had supposed herself a prime minister- in-waiting after Wednesday's resignation by Necmettin Erbakan, modern Turkey's first Islamist leader.
Mrs Ciller must now wait to see if Mr Yilmaz can conjure a majority from Turkey's fluid parliament.
If he cannot, the President is expected to ask Mrs Ciller to form a government.
Mr Yilmaz's task over the coming week is to cement the support already promised him by other opposition leaders and woo enough from Mrs Ciller's party to enable him to win a vote of confidence.
Mrs Ciller's miscalculation may cost her important support inside her party, which has not been in opposition since 1991. More important, her renowned infallibility appears to have been challenged.
Mr Demirel may have enjoyed his decision; he fell out with Mrs Ciller after she changed the True Path, which he founded. However, he knows he has the full support of Turkey's civil and military establishment. Mr Demirel was twice unseated by Turkey's generals. By asking Mr Yilmaz to form a government, some Turks feel he may have averted a fresh military intervention.
A mediocre prime minister on two brief occasions, Mr Yilmaz needs all his skills if he is to fend off what promises to be a determined assault from Mrs Ciller. He is by no means guaranteed success. If he fails, Mrs Ciller - along with her Islamist partners - will be waiting.