Royal Navy ship HMS Bulwark has already helped save the lives of more than 4,000 migrants - and tonight it will rescue more.
There are reports another 3,000 migrants have left Libya today, many on boats which are unseaworthy and do not even have engines.
After being towed out to sea, they are left to drift for days by human traffickers.
Defence Secretary Michael Fallon, who is aboard the ship tonight, told ITV News Europe is still not doing enough to tackle the problem.
He said: "The whole of Europe really need to adopt a more comprehensive approach to get after the traffickers, to understand how they're making money out of this trade in human misery and to do more to discourage people from leaving these countries."
Political correspondent Carl Dinnen reports from HMS Bulwark:
A British Navy warship has rescued 747 migrants trying to make the treacherous journey across the Mediterranean and delivered them to safety in Italy.
HMS Bulwark saved the migrants from dangerously overcrowded boats in the waters just north of Libya.
Many young children, some without parents, were among those rescued by crew aboard the ship.
ITV News Political Correspondent Carl Dinnen reports on their journey:
Hundreds of migrants fleeing war and poverty in Africa have been put ashore in Italy after a British navy ship rescued them from the Mediterranean. More than 700 people - including many children - were rescued by the crew of HMS Bulwark after their boat began to sink off Libya.
Hundreds of migrants - including 50 young children - have been rescued from an "unstable" ship in the Mediterranean sea near Libya, thanks to UK ship HMS Bulwark.
The double-decker, wooden-hulled boat was described by the Ministry of Defence as "heavily overcrowded", with 369 people packed on board.
As part of the international migrant search and rescue mission - codenamed Operation WEALD - Bulwark dispatched five of its specialised landing craft to rescue the migrants and bring them to safety.
All eight of its landing craft have been converted into rescue boats, piled high with lifejackets, medical equipment and emergency supplies.
The rescue was one of five which took place in the Mediterranean today, with Italian, Greek and Irish naval units all involved in saving some 800 lives. They will now be transferred to a dedicated port of safety.
Defence Secretary Michael Fallon welcomed news of the operation, but called for a better solution from UK and overseas governments.
HMS Bulwark and her crew have once again saved hundreds of lives in the Mediterranean migrant crisis, offering medical assistance, food, water and dry clothes to those in need.
A wider political solution is required to this crisis, but that does not detract from today’s rescue at sea.
Defence Secretary said Britain would send specialist staff to help develop the details of a mission to disrupt the criminal networks behind the flow of migrants heading to Europe via Libya.
Michael Fallon said: "We have made some progress today to a more comprehensive plan that will tackle the problem at source, disrupt the trafficker routes and get after the criminals who are involved in this activity.
"We have the Royal Navy already off the Libyan coast, helping in search and rescue.
"Any destruction of boats will require some legal authority and that has to come from the United Nations but we are not at that stage yet."
European Union foreign ministers have agreed to set up a naval mission in the Mediterranean to target gangs smuggling people from Libya to Europe.
The EU's foreign affairs chief Federica Mogherini had earlier said that an agreement would increase the chances of the United Nations Security Council giving the EU the backing many ministers want for a more aggressive mission.
The UN refugee agency says that 51,000 migrants have entered Europe by crossing the Mediterranean Sea this year, with 30,500 coming via Italy.
About 1,800 have drowned in the attempt.
EU officials have confirmed it will establish a "naval operation to disrupt the business model of smugglers and traffickers networks in Mediterranean".
Decision just taken to establish the EU naval operation to disrupt the business model of smugglers and traffickers networks in Mediterranean
The EU needs to work out a plan with the Libyan authorities to help stem the tide of migrants attempting to cross the Mediterranean, foreign secretary Philip Hammond has said.
Mr Hammond was speaking during a summit in Brussels aimed at finding a solution to the people-traffickers bringing desperate African and Middle Eastern migrants into Europe.
"There's quite a number of steps that have to be dealt with," he said. "But I think we're going to take an important first step in our decision today."