To the best of my understanding, current Israeli law doesn't really adequately provide for such circumstances as the incapacitation or death of the sitting PM - there is a provision within The Basic Law, under Government, which establishes that the Deputy PM shall fill in given the event of the sitting PM's temporary incapacitation, but the law leaves open who determines, or by what means determination is to be made, that the PM is unable to discharge his duties. There is no provision or proceedure for formal handover of power and responsibility. Deputy PM Olmert was advised by Sharon's doctors some hours after Sharon first underwent treatment for his just previous stroke that the PM would be unable to serve, and Olmert simply assumed the position without any formal authorization. The public wasn't even aware of the handoff for some days, and no official announcement was made for over a week following the event.
Meanwhile, at present, it appears from the news wires and from the streaming feed of Israel's Channel 2 News
(note - may be choppy or even inaccesible, current very heavy traffic) that the deathwatch may have begun.
Olmert has called on the Cabinet to convene in full at 9:00 AM. Sharon's doctors say " ... the operation is going well", but that " ... chances of recovery may not be high". One doctor, not a member of the surgical team working on Sharon, when asked directly about Sharon's prospects, responded "Let's be optimistic, some people survive it." An un-named "Close Associate" of Sharon is quoted as saying "Hope for a miracle". Though a hospital in Beersheeba was much nearer to Sharon's ranch when he was stricken, the decision was made by his personal physician, who was present, to transport him to the larger, more lavishly staffed and equipped Jerusalem hospital where he now is in surgery. Reportedly, his condition deteriorated during the transport, and was life-threateningly critical upon his arrival at the hospital.
Sharon's family is with him, and a service is being held at the West Wall, with attendance growing.
As Sharon's Kadima party essentially is without him nothing, and enjoys a huge lead in opinion polling leading up to the scheduled March 28 election, the political picture is far from clear. The Likud party might make a bid to set aside the election as an emergency measure, but Likud's frontman, Netanyahu, has said only that he prays for Sharon's health.