The killer launched his attack on soldiers from 1st Battalion The Royal Gurkha Rifles at a base in Helmand Province in southern Afghanistan early this morning.
He shot one dead in his sleeping quarters and killed the other two in the base's command centre using a shoulder-mounted rocket-propelled grenade launcher, sources said.
The Ministry of Defence described it as a "suspected premeditated attack". The soldiers' families have been informed.
A manhunt is under way after the murderer fled following the carnage at the joint patrol base near Helmand's provincial capital, Lashkar Gah, at about 2am local time.
Afghan president Hamid Karzai sent a letter of condolence to Britain offering his apologies for the incident, which came as a blow to the Nato mission to build up Afghanistan's police and army.
This is the second time members of Afghanistan's security forces have turned on UK troops with tragic results.
Five British soldiers were killed and six injured when an Afghan policeman opened fire on them at a secure checkpoint in Nad-e-Ali in Helmand in November.
Analysts said the Afghan soldier who carried out today's attack could have been a Taliban "sleeper" or may have been radicalised by insurgents after completing his training.
Lieutenant Colonel James Carr-Smith, spokesman for Task Force Helmand, said the killings were thought to be the actions of a "lone individual" who had betrayed his comrades.
"His whereabouts are currently unknown but we are making strenuous efforts to find him," he said.
"He should know that his actions will not deter us from our task and we will continue to work closely with our Afghan friends to bring security to Helmand.
"Three courageous and dedicated soldiers have made the ultimate sacrifice. They will be greatly missed and their actions will not be forgotten. We will remember them."
Lieutenant General Nick Parker, the British deputy commander of Nato forces in Afghanistan, admitted it was a "traumatic event".
But he said: "More widely across Helmand I think everybody knows, partnering is the key to our future, we have got to transfer security responsibility to the people whose country this is and if we don't do that, we're not going to succeed in our mission."
It is understood that two of the soldiers killed in the attack were British and the third was Nepalese.
Video: Rogue Afghan soldier kills three British
Prime Minister David Cameron condemned the "appalling" incident but said it should not change the strategy of working alongside Afghan forces.
He said: "The insurgents want us to change our approach, they want us to abandon our strategy, they want us to lose faith in the Afghan National Army. That would not be the right approach."
Defence Secretary Liam Fox added: "This is a despicable and cowardly act, and my thoughts go out to the family and friends of those who have lost their lives."
A massive programme to expand the Afghan security forces - the goal is that the army should grow from 85,000 in 2009 to 134,000 by 2011 - is under way, but concerns have been expressed about the quality and allegiance of some recruits.
The Nato-led International Security Assistance Force (Isaf) and the Afghan Ministry of Defence have launched a joint investigation into the killings.
Isaf commander General David Petraeus said: "I echo the condolences and sentiments offered by president Karzai and the other Afghan officials.
"This is a combined, joint mission - Afghan and alliance troopers fighting shoulder-to-shoulder against the Taliban and other extremists.
"We have sacrificed greatly together, and we must ensure that the trust between our forces remains solid in order to defeat our common enemies."
The incident had echoes of the shootings carried out by an Afghan policeman at a checkpoint in Nad-e-Ali in Helmand Province on November 3.
Warrant Officer Class 1 Darren Chant, 40, Sergeant Matthew Telford, 37, and Guardsman Jimmy Major, 18, from the Grenadier Guards, died alongside Corporal Steven Boote, 22, and Corporal Nicholas Webster-Smith, 24, from the Royal Military Police.
Another six British soldiers and two Afghan policemen were wounded.
The Taliban claimed responsibility for the killings but many in military circles believe the incident was unconnected to insurgent groups.
The British death toll in the Afghan campaign since 2001 now stands at 317.
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