The director, Shlomo Mor-Yosef, said Sharon remains in serious condition.
"The prime minister has had a CT scan that showed the cerebral hemorrhaging has stopped," Mor-Yosef said. "The prime minister is now being transferred to the neurosurgical intensive care unit. In the unit, he will be treated, observed and cared for."
"At this point, all the vital signs are ... stable. The prime minister is still in serious condition," the director said.
Powers were transferred to his deputy, Ehud Olmert.
Sharon fell ill at his ranch Wednesday evening and was rushed to Hadassah Hospital in Jerusalem, where he was diagnosed with a cerebral hemorrhage. Doctors began emergency surgery about midnight to stop "massive, wide-spread bleeding" in his brain.
Surgery apparently had been complicated by blood thinners he had been given following a mild stroke on Dec. 18, and the medication may also have contributed to Wednesday's stroke.
Mor-Yosef did not address Sharon's prognosis, but neurosurgeons not involved in Sharon's treatment said a full recovery was not likely following such a massive stroke. They said it usually takes at least a day after the surgery to determine the extent of any damage.
Olmert chaired a special Cabinet session Thursday. The session was convened to send a message of stability and continuity to the Israeli people, one government minister said.
A doctor told CBS News correspondent David Hawkins that people generally emerge from this type of surgery with severe disabilities.
Sharon's cerebral hemorrhage, or bleeding stroke, came at a time of upheaval among Palestinian factions in Gaza and in the midst of both Israeli and Palestinian election campaigns.
Hawkins reports that Sharon's stroke throws Israeli politics into a period of uncertainty as Sharon runs for re-election on March 28 as the head of a new centrist party, Kadima. He enjoys a wide lead in the polls. The party's strength is centered on Sharon, and if he were forced to step down, Israel's political scene would be thrown into turmoil.
Ehud Olmert, "is a veteran Israeli politician," said Daniel Kurtzer, former ambassador to Israel. "Although it was clear during the last months in the Likud [Party] that his popularity was beginning to wane. He is a proven leader but he will now have to build a popular base certainly for the next few months of governing."
In a written statement, President Bush praised Sharon as "a man of courage and peace," saying he and first lady Laura Bush "share the concerns of the Israeli people ... and we are praying for his recovery."
If Sharon is not able to continue his political relationship with Mr. Bush, it remains unclear whether his successor would have the clout and credibility to move the Middle East peace process forward, reports CBS News chief White House correspondent John Roberts.
Israeli Chief Rabbi Yona Metzger called on Israelis to read Psalms and pray for Sharon. "We are very, very worried," he said, and prayed for "mercy from Heaven."
Palestinian leader Mahmoud Abbas contacted Israeli officials Thursday to express concern over Sharon's health, Abbas' office said.
"President Abbas phoned Israeli officials over the health deterioration of Prime Minister Ariel Sharon and he voiced his concern about his health," said an Abbas aide, Nabil Abu Rdeineh.
Pan-Arab satellite television broadcasters beamed out largely straightforward, nonstop live coverage from outside the hospital where Sharon, a particularly despised figure among many Arabs, struggled for his life.
A radical Palestinian leader in Damascus, the Syrian capital, called the stroke a gift from God.
"We say it frankly that God is great and is able to exact revenge on this butcher. ... We thank God for this gift he presented to us on this new year," Ahmed Jibril, leader of the Syrian-backed faction Popular Front for the Liberation of Palestine-General Command, a small radical group, told The Associated Press.
Speaking to reporters outside the hospital, Sharon aide Raanan Gissin warned Israel's enemies: "To anyone who entertains any notion to try and exploit this situation ... the security forces and IDF (Israeli military) are ready for any kind of challenge," he said.
But a Palestinian commentator on the Saudi-owned Al-Arabiya network offered Sharon unexpected praise as "the first Israeli leader who stopped claiming Israel had a right to all of the Palestinians' land," a reference to Israel's recent withdrawal from the Gaza Strip.
"A live Sharon is better for the Palestinians now, despite all the crimes he has committed against us," said Ghazi al-Saadi.
Sharon's personal physician said early Thursday that he expects Sharon to emerge from surgery "safely."
"The prime minister is currently in surgery, it is proceeding properly," said Dr. Shlomo Segev. "We need to wait patiently. I expect him to emerge from it safely."
Doctors said chances of recovery were slim.
"It's among the most dangerous of all types of strokes," with half of victims dying within a month, said Dr. Robert A. Felberg, a neurologist at Ochsner Clinic in New Orleans.
"The fact that he's on a respirator means it's extremely serious," said Dr. Philip Steig, chair of neurosurgery at Weill-Cornell Medical Center in New York.