Ehud Olmert, leader of the centrist Kadima party which Sharon founded late last year, has officially become prime minister.
Sharon, 78, suffered a stroke on January 4 and has been comatose ever since, despite many operations.
Under Israeli law, an ailing prime minister can have a temporary replacement for no more than 100 days before an official successor must be named.
The cabinet vote was held early because Israel's week-long Passover holiday begins on Wednesday.
Tuesday's declaration was largely symbolic, as Olmert has been Israel's acting leader since January and medical experts believe Sharon's chances of recovery to be remote.
Kadima, which came first in elections held last month in the country, is currently forming a coalition government.
First elected prime minister in 2001, Sharon oversaw the controversial withdrawal of Israeli soldiers and settlers from the Gaza Strip last year.
This caused dissent within his Likud party and led him to form Kadima.
Officials at Hadassah Hospital in Jerusalem, where Sharon is being treated, told AP that discussions were still under way on whether to move Sharon to a long-term care facility.