Mr Olmert, who was made acting leader three months ago when his mentor and predecessor Ariel Sharon suffered a massive stroke, led Kadima to a narrow win in last week's election but has won the endorsement of enough rival parties to establish a new coalition.
"I am honoured to announce that I have decided to entrust Ehud Olmert with the task of forming the next government," President Katsav told a news conference after talks with the Kadima party leader at his official residence in Jerusalem.
"I praise Olmert's announcement that he seeks to form a stable and broad government and that he has not rejected any party from joining it."
Mr Katsav said a clear majority of newly elected MPs had recommended Olmert be given the job following talks with representatives of the 12 political parties who won seats in the 120-member parliament.
"During my meetings with the different factions, 78 MPs recommended that I appoint Mr Olmert. No other name was put forward."
Coalition talks
Mr Olmert pledged to complete the coalition talks as quickly as possible but said it would take time to ensure a line-up that could last until the scheduled end of his term of office in 2010.
"I will do my utmost to complete this task as quickly as possible," said
Mr Olmert as he sat alongside Mr Katsav.
Mr Olmert now has 28 days to form a cabinet, with a possible 14-day extension, before the line-up is put to a parliamentary vote of confidence.
He has pledged to use his mandate to set the final borders of Israel by 2010, with or without agreement from the Palestinians, by withdrawing from parts of the occupied West Bank and retaining key settlement blocs.
Labour leader Amir Peretz has declared he will join the coalition and is likely to become defence minister.
Mr Olmert is also said to be supported by at least the ultra-orthodox Shas party, with 12 seats, the Pensioners, with seven and the religious United Torah Judaism on six.
Commentators suggest a deal is also close with the ultra-nationalist Yisrael Beitenu party, which advocates the forced deportation of Arab Israelis in exchange for Jews being evicted from the West Bank.