The party's election on Monday of Olmert as its acting chief means the acting prime minister is to lead Kadima into 28 March parliamentary elections.
If election results conform with poll previews and Kadima wins the most seats in parliament, then Kadima would form the next government, and Olmert, as party leader, would be prime minister.
On Sunday, Attorney General Meni Mazuz ruled that Olmert would remain prime minister through the March balloting.
Sharon walked out of Likud Party and formed Kadima two months ago after concluding he would waste too much energy doing battle with Likud lawmakers who oppose territorial concessions to the Palestinians.
Leading Likud members flocked after him into his new centrist faction, which wants to draw Israel's final borders, with or without the Palestinians.
Several high-profile Labour Party leaders joined the new party, too, most notably Nobel Peace Prize laureate Shimon Peres. Polls showed the party taking more than one-third of parliament's 120 seats.
Then, abruptly, Sharon suffered a stroke and went into a coma. One of Sharon's neurosurgeons, Dr Jose Cohen, said shortly afterwards that Sharon would not return to power.
His departure from the political stage had fuelled speculation that Kadima would collapse without him. But polls so far have not borne that out. A survey of 501 people by the Dahaf agency published on Friday showed Kadima winning 42 seats, as opposed to 17 and 13, respectively, for Likud and Labour.
Olmert, meanwhile, has quietly taken charge. On Sunday, he steered cabinet through a controversial decision to allow Palestinians to vote in east Jerusalem for their parliament, and warned of harsh action against Jewish settlers who have been rioting in a volatile West Bank city.
In the past four years, Olmert had been Sharon's point man, especially on Israel's recent withdrawal of Jewish settlers and soldiers from the Gaza Strip.