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.
Dr. Larry Goldstein, director of Duke University's stroke program, said much depends on the extent, location and duration of the bleeding.
"Bleeding in some areas of the brain, if it's caught early enough, you can actually have not a bad outcome," he said.
Sharon was put in an ambulance at his ranch in the Negev Desert after complaining about feeling unwell. The stroke happened during the hourlong drive to Hadassah Hospital in Jerusalem, Dr. Shmuel Shapira of the hospital told Channel 10 TV.
Doctors checking Sharon late last month said he weighed 118 kilograms (260 pounds) at the time of the first stroke but had since lost more than 2 1/2 kilograms (six pounds) and was otherwise in good health. Sharon is about 170 centimeters (5-foot-7).
The prime minister had been taking blood thinners since the first stroke to prevent another clot, but such drugs also raise the risk of cerebral hemorrhages, which account for only about 10 percent of strokes. Other possible causes are ruptured blood vessels, an aneurysm, or bulge in a vessel wall that bursts, or even chronic high blood pressure.
Security agents and police spread out around the Jerusalem hospital before Sharon arrived, setting up a security perimeter. Later, they surrounded Olmert's residence in Jerusalem.
Cabinet Secretary Yisrael Maimon said Sharon's authority was transferred to Olmert because the prime minister was under general anesthesia. Under Israeli law, he will serve as acting prime minister until Sharon can resume his powers.
On Dec. 18, Sharon was taken to Hadassah Hospital from his office after suffering the mild stroke. Doctors said he would not suffer long-term effects, but they discovered a birth defect in his heart that apparently contributed to the stroke.
Sharon had been scheduled to check into the same hospital Thursday for a procedure to repair a tiny hole between the upper chambers of his heart. Doctors said the blood clot that briefly lodged in Sharon's brain last month, causing the mild stroke, made its way through the hole and from there to a cranial artery.
Sharon first came to prominence as an army officer, setting up a unit that fought Palestinian infiltrators in the 1950s. Advancing through the ranks of the army, he served as commander of the Gaza region after Israel captured the territory in the 1967 war, launching punishing raids.
After serving in the 1973 Mideast war, Sharon left the military and entered politics, forging the hardline Likud Party, which came to power in 1977.
As defense minister, he directed Israel's ill-fated invasion of Lebanon in 1982 and was forced to step down by an Israeli commission of inquiry, which found him indirectly responsible for a massacre of Palestinians in two refugee camps by Christian Phalangist soldiers.
Sharon re-emerged as prime minister in 2001, and two years later he reversed his course of decades of support for Jewish settlement construction and expansion in the West Bank and Gaza, promoting a plan for unilateral withdrawal from Gaza and part of the West Bank. The pullout was completed in September.
The withdrawal fractured his Likud party, and he left it to form Kadima with a platform of seeking a compromise for peace with the Palestinians. He was putting together a list of candidates for the parliamentary election when he fell ill Wednesday.
In the March 28 election, Sharon had been expected to face off against Benjamim Netanyahu, the tough-talking former prime minister who recently won the Likud primaries, and Amir Peretz, the union leader who recently unseated veteran Israeli politician Shimon Peres as head of the liberal Labor Party.
Olmert, who could emerge as Sharon's successor in Kadima, would likely have a far tougher time beating either Netanyahu or Peretz than Sharon would have.
According to Israeli law, Olmert as deputy premier assumes the post of prime minister for 100 days if Sharon becomes incapacitated. Then, Israel's ceremonial president would meet with political leaders and choose someone to form a coalition government.
Two senior U.S. envoys postponed their visit to Israel and the Palestinian areas because of Sharon's condition, Palestinian officials said Thursday.
U.S. National Security Council official Elliott Abrams and State Department official David Welch were to have arrived in Israel later Thursday, to try to settle a dispute between Israelis and Palestinians over the participation of Jerusalem residents in Palestinian parliament elections Jan. 25.