Mr Olmert, a recent election winner, was Mr Sharon’s deputy and named interim prime minister when the leader suffered a massive stroke on January 4.
Under Israeli law, Mr Sharon was declared permanently incapacitated and unable to serve 100 days after suffering his stroke.
At a special session, the cabinet designated Mr Olmert acting prime minister, an appointment which Cabinet Secretary Yisrael Maimon said would go into effect on Friday.
Mr Olmert, leader of the centrist Kadima party that Mr Sharon founded last November, is currently putting together a new coalition government.
The new administration will be tasked with setting the final borders of the Jewish state by pulling tens of thousands of settlers out of the occupied West Bank, following on from Mr Sharon’s decision to uproot Jews from the Gaza Strip last summer.
Mr Olmert, 60, is expected to become prime minister in his own right once the new government is sworn in.
“I hope today’s decision will only be a short-term one,” he told his ministers after the cabinet decision.
“We are busy forming the next government. I hope we will manage to finish the task quickly.
The government decision was moved forward due to the week-long Jewish Passover holiday which starts on Wednesday.
“This is a very difficult and sad day for all of us. We never imagined we would reach this moment. We offer our prayers and best wishes to the prime minister, his family and his friends for his recovery,” said Cabinet minister Israel Maimon.
Mr Sharon, who became premier in early 2001, dominated the Israeli political scene until he suffered his brain haemorrhage, and had been expected to lead the newly formed Kadima party to victory in a March 28 election.
Doctors initially induced him into a coma but efforts to rouse him over the last three months have been in vain despite initial responses to pain stimulus tests.
A spokesman for Jerusalem’s Hadassah hospital, Yael Bossam-Levy, told AFP on Tuesday that Mr Sharon remains “serious but stable”.
Mr Sharon gained a reputation as an arch hawk when he initially assumed the leadership, however in his second term did an about-face, pulling Israeli settlers and soldiers out of the occupied Gaza Strip last year.
Had he been re-elected, it was widely expected that Mr Sharon would embark on a campaign to fix the final borders of the Jewish state — a task that will not fall onto Mr Olmert’s shoulders.