The military assault began shortly before sundown, with one helicopter skimming very close to the roof of the shopping complex as a loud explosion rang out, far larger than any previous grenade blast or gunfire volley.
Kenyan police said on Twitter that security forces had launched a "MAJOR" assault to end the bloody siege.
"This will end tonight. Our forces will prevail. Kenyans are standing firm against aggression, and we will win," Kenya's National Disaster Operation Centre said on Twitter.
The Kenya Defense Forces later said it had rescued "most" hostages and had taken control of most of the mall, though it did not provide details.
Many of the rescued hostages -- mostly adults -- were suffering from dehydration, Col. Cyrus Oguna, a military spokesman, told The Associated Press. He refused to say how many hostages were rescued or how many were still being held. He said some of the attackers had "most probably" been killed in the operation.
The assault came about 30 hours after 10 to 15 al-Shabab extremists stormed the mall Saturday from two sides, throwing grenades and firing on civilians.
Loud exchanges of gunfire rang out from inside the four-story mall throughout Sunday. Kenyan troops were seen carrying in at least two rocket-propelled grenades. Al-Shabab militants reacted angrily to the helicopters on Twitter and warned that the Kenyan military action was endangering hostages.
Kenyan authorities said they would do their utmost to save hostages' lives, but no officials could say precisely how many people were being held captive. Kenya's Red Cross said in a statement, citing police, that 49 people had been reported missing. Officials did not make an explicit link but that number could give an indication of the number of people held captive.
Kenya's Red Cross said the death toll rose to 68 after nine bodies were recovered Sunday. More than 175 people were injured, including many children, Kenyan officials said.
Somalia's al-Qaida-linked rebel group, al-Shabab, claimed responsibility for the attack that specifically targeted non-Muslims, saying it was in retribution for Kenyan forces' 2011 push into neighboring Somalia.
U.S. Secretary of State John Kerry condemned what he called "an enormous offense against everybody's sense of right and wrong," and called the attackers "ruthless and completely reckless terrorists."
Kerry, who was in New York, for meetings at the United Nations, spoke Sunday with Somalia's foreign minister and U.N. ambassador.
State Department spokeswoman Marie Harf said U.S. law enforcement, military and civilian personnel in Nairobi were providing advice and assistance to the Kenyan authorities. She said five Americans were among the scores of people injured in the attack, but the U.S. had no reports of any American deaths.
Earlier in the day, al-Shabab said on its new Twitter feed -- after its previous one was shut down Saturday -- that Kenyan officials were asking the hostage-takers to negotiate and offering incentives.
"We'll not negotiate with the Kenyan govt as long as its forces are invading our country, so reap the bitter fruits of your harvest," al-Shabab said in a tweet.
Kenyan President Uhuru Kenyatta, who lost a nephew and the nephew's fiancee in the attack, reiterated his government's determination to continue fighting al-Shabab.
"We went as a nation into Somalia to help stabilize the country and most importantly to fight terror that had been unleashed on Kenya and the world," said Kenyatta. "We shall not relent on the war on terror."
Although this violent attack had succeeded, the Kenyan leader said, the country's security forces had "neutralized" many others.
Former Kenyan Prime Minister Raila Odinga told reporters that "quite a number" of people were being held hostage in two areas of the sprawling complex, which includes stores for such retail giants as Nike, Adidas and Bose. Many hostages were believed to be in a grocery and general department store called Nakumatt.
Kenyan security officials sought to reassure the families of hostages but implied that some of those being held could be killed.
"The priority is to save as many lives as possible," said Interior Cabinet Secretary Joseph Lenku, adding that more than 1,000 people escaped the attack inside the mall on Saturday.
"We have received a lot of messages from friendly countries, but for now it remains our operation," Lenku said, adding that Kenyan forces controlled the mall's security cameras.
Westgate Mall is at least partially owned by Israelis, and reports circulated that Israeli commandos were on the ground to assist in the response. Four restaurants inside the mall are Israeli-run or owned.
In Israel, a senior defense official said there were no Israeli forces participating in an assault, but said it was possible that Israeli advisers were providing assistance. The official, speaking on condition of anonymity because he was discussing a classified military issue, would not elaborate.
Israel has close ties to Kenya going back many years. In recent years, Israel has identified East Africa as an area of strategic interest and stepped up ties with Kenya and other neighboring countries, due to shared threats posed by al-Qaida and other extremist elements. In 2002, militants bombed an Israeli-owned luxury hotel near Mombasa, killing 13 people, and tried to shoot down an Israeli airliner at the same time.
Kenyans and foreigners were among those confirmed dead, including British, French, Canadians, Indians, a Ghanaian, a South African and a Chinese woman.
Britain's prime minister, in confirming the deaths of three British nationals, told the country to "prepare ourselves for further bad news."
Kofi Awoonor, a Ghanaian poet, professor and former ambassador to Brazil, Cuba and the United Nations, died after being wounded in the attack, Ghana's presidential office confirmed. Ghana's ministry of information said Awoonor's son was injured and is responding to treatment.
Kenya's presidential office said that one of the attackers was arrested on Saturday and died after suffering from bullet wounds.
Britain's Foreign Office said that Foreign Secretary William Hague chaired a meeting of Britain's crisis committee and sent a rapid deployment team from London to Nairobi to provide extra consular support.
The United Nations Security Council condemned the attacks and "expressed their solidarity with the people and government of Kenya" in a statement.
There was some good news on Sunday, as Kenyan media reported that several people in hiding in the mall escaped to safety in the morning, suggesting that not everyone who was inside overnight was being held by al-Shabab.
Police lobbed multiple rounds of tear gas throughout the day to disperse hundreds of curious Kenyans who gathered near the mall.
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