Last Updated 10:55 a.m. ET
CAIRO - Egypt's military threw its weight Friday behind President Hosni Mubarak's plan to stay in office through September elections, while protesters fanned out to the presidential palace in Cairo, the state-run TV building and other key symbols of the authoritarian regime in a new push to force the leader to step down immediately.
The protests continue as President Mubarak temporarily left Cairo for Sharm el-Sheikh in southern Egypt, sources told CBS News.
There were reports that Mubarak had left the country for medical care, but they are inaccurate, the sources said. State-run TV is also saying that there will be an announcement from the president's office shortly.
The Armed Forces Supreme Council's statement of support for Mubarak - its second in two days - was a blow to many protesters who had called on the military to take action to push out Mubarak after his latest refusal to step down.
But soldiers also took no action to stop demonstrators from massing outside the palace and the headquarters of state television, indicating they were trying to avoid another outbreak of violence.
Anti-government protesters said they were more determined than ever as the uprising entered its 18th day.
"We expected the army's decision, we always knew that it was behind Mubarak. But we know it's not going to harm us," Safi Massoud said as she joined thousands of people packed into Cairo's central Tahrir Square. "We won't leave until we choose a transition president. We don't want Mubarak, we don't want Suleiman."
CBS News correspondent Terry McCarthy reported that "some of the biggest crowds Cairo has ever seen" were gathering in Tahrir Square, the focal point for the nearly three-week protest movement.
In nearby Heliopolis, CBS News' Amjad Tadros saw hundreds of protesters around the presidential palace. The palace was protected by four tanks and rolls of barbed wire, but soldiers were doing nothing to stop demonstrators from joining the rally and chanting anti-Mubarak slogans.
There were, however, several reports from Arab networks that clashes had broken out between anti-government protesters and Mubarak supporters.
The military statement endorsed Mubarak's plan to transfer some powers to Vice President Omar Suleiman and promised free and fair presidential elections later this year.
It also promised that the hated emergency laws, in force since Egypt's authoritarian ruler came to office in 1981, would be lifted and gave a somewhat more specific timeframe than Mubarak had offered in his Thursday night speech.
The military implied they would be lifted when protests end, saying it could happen "when the current security situation permits."
It also called for public services to resume and urged "the return of normal life in order to safeguard the achievements of our glorious people."
CBS News national security correspondent David Martin reports, however, that the powerful military's unwillingness to act decisively against Mubarak on Thursday spoke for itself.
"Essentially, they decided to stick with Mubarak, which is what they've done since the beginning of this crisis. They have remained loyal to a regime which treated them very well for 30 years," Martin told CBS Evening News anchor Katie Couric.
"The Egyptian military is not interested in democratic reform. It's interested in stability. It just doesn't want to pay too high a price for that stability. So it doesn't want to ruin its reputation by using violence against the protesters, and it doesn't want to tear its relationship with the U.S. military, which would cost it about a third of its defense budget," added Martin. "So if the protesters remain out in the streets, the Egyptian military has some very tough choices to make."
Other protesters massed outside the Cabinet, parliament and the state television headquarters several blocks away from Tahrir Square.
Hundreds of demonstrators formed a human barricade around the building that houses state TV and radio, checking IDs and turning away those who work there. Tanks and barbed wire surrounded the building overlooking the Nile, but troops did not keep protesters away.
In Alexandria on the Mediterranean coast, hundreds of thousands took to the streets after Friday prayers.
Delivering a sermon in Alexandria's main mosque, Sheikh Ahmed al-Mahalawi told worshippers they were bringing down a "corrupt regime" that was not fit to govern, and told them not to back down.
Protesters took over the seafront streets, marching near the presidential palace there.
Hopes that Mubarak would resign had been raised Thursday when a council of the military's top generals announced it had stepped in to secure the country, and a senior commander told protesters in Tahrir Square that all their demands would soon be met.
Instead, several hundred thousand people watched in disbelief and anger as Mubarak refused to step down.
Mubarak called the protesters' demands legitimate and promised that September presidential elections -- in which he says he will not run -- will be "free and fair" with supervision to ensure transparency.
He said that on the recommendation of the panel, he had requested the amendment of five articles of the constitution to loosen the now restrictive conditions on who can run for president, to restore judicial supervision of elections, and to impose term limits on the presidency.
He also annulled a constitutional article that gives the president the right to order a military trial for civilians accused of terrorism. He said that step would "clear the way" for eventually scrapping the emergency law but with a major caveat -- "once security and stability are restored."
The emergency law gives police virtually unlimited powers of arrest.
"I think it's increasingly clear that this is a false transition. It's a transition where a game of musical chairs where they put people in different positions, but the same people - the same people who've been ruling Egypt for 30 years," Nicholas Burns, former undersecretary of State during the Bush administration, told CBS' "The Early Show." "And I don't think that handing off some powers by President Mubarak to Vice President Suleiman is going to placate the demonstrators and protesters all throughout Egypt."
Prominent reform advocate Mohamed ElBaradei, whose supporters were among the organizers of the 18-day-old wave of protests, called in a Twitter message on the army to step in "to rescue Egypt," warning the country might "explode."
In, ElBaradei said it would be "absurd" for the United States and other governments to continue to give backing to the Mubarak regime.
Another leading figure of the protest movement, Google executive Wael Ghonim, called for caution.
"The situation is complicated. I don't want the blood of the martyrs to be wasted and at the same time I don't want to see more bloodshed," he said in comments posted on Facebook.
The Muslim Brotherhood, Egypt's largest and best organized opposition group, called the speech a "farce."
"This is an illegitimate president handing power to an illegitimate vice president," said Mohammed Abbas, who represents the Brotherhood's youth wing. "We reject this speech and we call on Mubarak to step down and hand his powers to the army."
In his address on state TV, Mubarak showed the strategy he has followed throughout the days of upheaval, trying to defuse the greatest challenge ever to his nearly three-decade authoritarian rule. So far, he has made a series of largely superficial concessions while resolutely sticking to his refusal to step down immediately or allow steps that would undermine the grip of his regime.
Looking frail but speaking in a determined voice, Mubarak spoke as if he were still in charge, saying he was "adamant to continue to shoulder my responsibility to protect the constitution and safeguard the interests of the people." He vowed that he would remain in the country and said he was addressing the youth in Tahrir as "the president of the republic."
Even after delegating authority to his vice president, Mubarak retains his powers to request constitutional amendments and dissolve parliament or the Cabinet. The constitution allows the president to transfer his other authorities if he is unable to carry out his duties "due to any temporary obstacle."
"I saw fit to delegate the authorities of the president to the vice president, as dictated in the constitution," he said.
President Barack Obama appeared dismayed by Mubarak's announcement. He said in a statement that it was not clear that an "immediate, meaningful" transition to democracy was taking place and warned that too many Egyptians are not convinced that the government is serious about making genuine change.
Suleiman was already leading the regime's efforts to deal with the crisis, though he has failed to ease the protests, which have only escalated in size and ambition, drawing crowds of up to a quarter-million people.
Suleiman has also offered dialogue with the protesters and opposition over the nature of reforms. In a sign that he is sticking to that strategy, state TV reported that he asked Prime Minister Ahmed Shafiq to appoint a deputy prime minister to be in charge of the dialogue with the protesters and the opposition.
Despite the overwhelming sense of disappointment among the protesters, some noted that Mubarak's immediate resignation would have had unintended consequences. His immediate departure would have triggered presidential elections within 60 days, with most of the restrictions that prevented free voting in the past still in place, said Amr Hamzawy, an Egyptian legal expert.
By transferring most powers to Suleiman and initiating constitutional amendments, Mubarak did the maximum possible under the constitution to meet the demands of the protesters, Hamzawy said.
"He went in a direction that is more preferable to open Egyptian politics in the next few months," said Hamzawy.
Despite those concerns, Denmark's prime minister, Lars Loekke Rasmussen, became the first Western leader to publicly call for Mubarak's resigntion.
"Mubarak is history, Mubarak must step down," Rasmussen said Friday in Copenhagen.