It is the biggest single loss of UK life at one time in Afghanistan since a Nimrod crash killed 14 in 2006.
The explosion is thought to have been caused by a very large Taliban bomb.
The number of British military deaths in Afghanistan since 2001 is now 404.
AnalysisThis is the worst single loss of life in one incident due to enemy action for UK troops in Afghanistan since the start of the war in 2001.
The last time a similar number of UK casualties was caused by improvised explosive devices (IEDs) was in the summer of 2009, when five British soldiers died in an IED blast.
The device remains the deadliest weapon in the Taliban's arsenal. It is effective on a psychological level, as well as causing deaths and severe injury, and is relatively cheap and simple to manufacture.
The level of suspected casualties suggests a large amount of explosives were used.
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In a statement, the Ministry of Defence (MoD) said the six soldiers were on a security patrol in a Warrior armoured fighting vehicle when it was caught in an explosion in Kandahar province.
Brig Patrick Sanders, commander of Task Force Helmand, said it suffered "catastrophic damage".
They had been travelling as part of a two Warrior patrol when the vehicle was hit at the junction of a road travelling east from Gereshk and another heading north to Lashkar Gah, the MoD said.
The BBC's Orla Guerin, in the Afghan capital Kabul, said there must have been a "massive explosion" to damage such a robust and heavily protected vehicle.
The area, she said, was sparsely populated and particularly unstable, according to Afghan officials, and insurgents were known to have planted roadside bombs there.
Meanwhile, BBC defence correspondent Caroline Wyatt said the six soldiers had been in the country for less than a month, having flown out of the UK on Valentine's Day.
She added that the ongoing operation to recover the bodies would have been slowed down as rescuers would have faced the risk of being injured themselves by further bombs.
News emerged earlier on Wednesday that the six were missing, presumed dead. It is understood the MoD has not formally identified them.
Most of the 9,500 UK troops in Afghanistan are expected to be withdrawn by the end of 2014, when 13 years of combat operations in the country are set to cease.
Defence Secretary Philip Hammond said the timetable for withdrawal remained on track despite this "cowardly attack".
"This will not shake our resolve to see through the mission - I believe we owe that to all the brave men and women who have sacrificed their lives and put themselves at risk over the last few years," he told the BBC.
He said a clear message had to be sent to the Taliban that - whether UK troops or Afghan troops were there - they "will not win on the battlefield - they never win on the battlefield".
Mr Cameron said he had spoken to the UK's leading military officers, who had stressed the commitment of troops to "getting the job done".
He also said he would discuss the Afghanistan situation with President Barack Obama on his visit to the US next week to ensure they were "in lock step" about the importance of training up the Afghan army and police, and making sure all Nato partners had a properly co-ordinated process for transition.
Labour leader Ed Miliband expressed his "profound sadness" over the latest deaths and said the UK owed those who put themselves in harm's way for our benefit "an immense debt of gratitude".
Chief of Defence Staff General Sir David Richards said he was "deeply saddened" by the news.
"The courage, fortitude and determination of those servicemen and women currently in Helmand is inspirational," he said.
The Archbishop of York John Sentamu said his heart went out to those affected, and we owed a debt of gratitude to our "brave soldiers".
The last British soldier to be killed in Afghanistan was Senior Aircraftman Ryan Tomlin, of 2 Squadron, Royal Air Force Regiment, who died from gunshot wounds in Helmand on 13 February.
Warrior armoured vehicle
|Source: British Army|
|Crew:||Three, plus seven fully equipped soldiers|
|Speed:||Max speed of 75km/h, average speed 40km/h|
|Weaponry:||Turret-mounted cannon, anti-helicopter chain gun and eight portable Light Anti-Armour (LAW) or anti-tank weapons|
|Armour:||Shielding on front, sides and underneath gives extra protection|