Since the newspaper's are wrongly reporting that Kadima members just chose the next Prime Minister, let me clarify the law.

First, so long as Olmert doesn't resign, he remains PM.

Second, if he does resign, there begins a long process of replacing him, during all of which time he remains caretaker PM. (If, in addition to resigning, he also declares himself unable to continue even as caretaker, he is replaced by the Deputy PM -- who happens to be Livni, independently of her victory in the primary. This is extremely unlikely.)

Upon his resignation, the President assigns the opportunity to put together a coalition to the candidate most likely to succeed in doing so. It should be emphasized that, unlike what has been written in countless editorials, the President is not bound to give the first opportunity to Livni. In fact, on one previous occasion (in 1990), Yitzhak Shamir had the best chance to form a coalition after a vote of no-confidence in Shamir's government, but then-President Chaim Herzog, gave the first shot to another candidate -- Shimon Peres. (In what has come to be known as "hatargil hamasriach", Peres failed to form a coalition and then Shamir succeeded. Anyone with lingering sympathy for Aryeh Deri, should pay careful attention to his role in that fiasco.)

Nevertheless, since Livni has made all the right elitist lefty noises, Peres will almost certainly give her the first chance. This is appropriate, given how well Livni's squeeky-clean image plays in, for example, the village of Baka Jat, where she took 172 out of 186 votes. Seems they just love her there.

If Livni fails to put together a coalition within 42 days, Peres can choose another candidate. This process can, in principle, continue until the elections scheduled for 2010, during all of which time Olmert remains caretaker PM. By the way, at any time during this process 61 MKs who agree on a single candidate can force the issue.

Likewise, at any time -- before or after a coalition is formed -- 61 MKs can vote to go to new elections within 90 days. This is very likely to happen.

Addendum: While in the original Basic Law: The Government, the process of choosing a candidate to form a government is iterative, this is no longer the case . As Anonymous comments, under the current version of the law, there can be at most two rounds. On a practical level, this doesn't make much difference since it is highly unlikely that there would ever be more than two candidates with any chance at all of forming a government. On a theoretical level, however, the situation is actually worse than I put it. Since, the sitting PM remains in office until a new government is formed, Olmert could in principle remain PM so long as he lives and lies. Even after elections, so long as nobody succeeds at obtaining the Knesset's approval, the sitting PM continues.