* Run-off to be held on June 16-17
* Vote between Islamist, military man divides nation
* Shafiq plays on fears about social freedoms
* Mursi says Shafiq seeks to rebuild Mubarak's state
By Yasmine Saleh and Edmund Blair
CAIRO, June 3 (Reuters) - Egyptian presidential candidate Ahmed Shafiq stepped up his attacks against his Muslim Brotherhood rival in a run-off vote, saying on Sunday the Islamist would drag Egypt into the "dark ages" and threaten the rights of women, Christians and others.
The comments from the last Prime Minister of former president Hosni Mubarack, his first public address for more than a week, reflected how divisive the race has become since the former air force commander made it through last month's first round into a head-to-head against Mohamed Mursi.
For the bloc of Egyptians who voted for centrist candidates in last month's first round, the outcome could hardly have been worse. Many of them worry as much about putting a conservative Islamist in charge as they do about handing power back to a man who, like Mubarak, has a military background. Mubarak was jailed for life on Saturday.
The run-off will be on June 16 and 17. The lead-up to the poll has been marred by violence, in which several of Shafiq's campaign offices have been attacked, and many protesters have hit the streets to demonstrate against both candidates.
"I represent a civil state, the Brotherhood represents a sectarian Brotherhood state. I represent moving forward, they represent going backwards," Shafiq said.
Sunday's statement appeared mainly to play on the fears of liberals, Christians (who make up a tenth of Egypt's 82 million people) and women. "Women of Egypt, I will not permit that the powers of extremism take you back to the dark ages," he said.
Shafiq, 70, who delivered his statement in a five-star hotel on Cairo's outskirts where there was a strong police presence, said the Brotherhood were "liars" and wanted to "penetrate" all institutions to create a state in line with their views.
He accused the Brotherhood of making a power grab despite promising not to run for the presidency, after the organisation also won more seats in a parliamentary vote than it originally said it would seek.
He suggested Mursi would answer to a religious leader not the people and would create sectarian divisions.
Shafiq had criticised the Brotherhood on May 26 at a news conference, after initial results showed he was in the run-off. But Sunday's broadside was his fiercest in the campaign so far. It was also timed to coincide with the start of overseas voting.
Mursi, 60, in his news conference on Saturday, had presented himself as the "revolutionary" candidate in the race and said his rival would rebuild Mubarak's old regime.
Mursi has said he would quit the Brotherhood and its party if elected and insists Christians would enjoy the same rights as everyone under his presidency, while women would be free to wear what they want, with no obligation to put on a Muslim veil.
Mursi has sought to reach out to more centrist candidates who lost in the first round, namely ex-Brotherhood member Abdel Moneim Abol Fotouh and leftist Hamdeen Sabahy. But he has yet to win explicit backing from either of them.
But those two losing candidates have been vociferous opponents of the "feloul", a derisive Arabic term referring to "remnants" of Mubarak's old order, such as Shafiq. They have vowed to protect the gains achieved since the uprising that toppled Mubarak on Feb. 11, 2011.
Sabahy and Abol Fotouh, who came third and fourth in the first round, secured a total of about 40 percent of votes cast, compared with less than a quarter won by each of Mursi and Shafiq. That means the centre-ground voters could be vital for victory in the run-off.
Turnout was 46 percent of around 50 million eligible voters.
Shafiq has not sought meetings with the other two candidates, a move that would certainly struggle given their opposition to "feloul", but his news conference on Saturday, in which he took no questions, was clearly aimed at picking off voters in the centre and those with misgivings about the Brotherhood.
"Choose a president of Egypt who will make a country for everyone, and not for one sect," he said, adding he was reaching out to every political power "even if it disagrees" with him.
Though he does not apologise for his links to Mubarak, who he has described as a role model, Shafiq insisted he would not rebuild the old order. He said there could be no return to jailing people for their views and said he would stamp out corruption, among the issues that drove the anti-Mubarak revolt.
One of his strongest cards is that, as a former military man, he is seen by many Egyptians as having the army's backing to help restore order after almost 16 months of turmoil.
"We will work immediately to restore security," he said.
He repeated another of his regular themes, that he has experience of office, and also promised poor farmers he would ensure fair prices for their produce, cancel debts they have with a state bank and give them health insurance.
To business, he pledged to draw in foreign investors, scared away by the turbulence since Mubarak's ouster. (Editing by David Holmes)