Lord Hall said he now accepts the BBC should have taken a different approach.
On Saturday, BBC Radio 1Xtra DJ Sideman - real name David Whitely - quit the station over the BBC's use of the N-word.
He said "the action and the defence of the action feels like a slap in the face of our community".
His actions were backed by a number of politicians and BBC staff, who offered support to the DJ.
In an email, sent to all BBC staff, Lord Hall said: "I recognise that we have ended up creating distress amongst many people."
His statement followed high-level discussions with BBC colleagues on Sunday morning.
'Strengthening guidance'
The Points West story described an attack on a 21-year-old NHS worker and musician known as K or K-Dogg, who was hit by a car on 22 July while walking to a bus stop from his workplace, Southmead Hospital in Bristol.
In his message, Lord Hall emphasised it was "the BBC's intention was to highlight an alleged racist attack".
"This is important journalism which the BBC should be reporting on and we will continue to do so," he said. "Yet despite these good intentions, I recognise that we have ended up creating distress amongst many people.
"The BBC now accepts that we should have taken a different approach at the time of broadcast and we are very sorry for that. We will now be strengthening our guidance on offensive language across our output.
"Every organisation should be able to acknowledge when it has made a mistake. We made one here."
K-Dogg suffered serious injuries including a broken leg, nose and cheekbone in the attack.
Police said the incident is being treated as racially aggravated due to the racist language used by the occupants of the car.
In its initial defence, the BBC said that the organisation felt it needed "to explain, and report, not just the injuries but, given their alleged extreme nature, the words alleged to have been used".
It said at the time that the decision, which was supported by K-Dogg's family, had not been taken lightly and that the BBC understood people would be upset.
A political row between the UK and France over migrant crossings in the Channel has intensified, with Britain's immigration minister saying the "unacceptable" increase in numbers "simply cannot go on".
Chris Philp is due to meet his French counterpart in Paris on Tuesday to discuss stronger measures, including "interceptions and returns".
It comes as 151 migrants reached the UK on Saturday - taking the total since 1 January to 4,100.
While southeast England basked in roasting sunshine on Sunday, a dozen people arrived in Dover after making the dangerous crossing.
Migrants met by Border Force in Kent
Mr Philp said the UK government wants to make the route across the Channel "completely unviable" so migrants "have no incentive to come to northern France or attempt the crossing in the first place".
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"We intend to return as many migrants who have arrived as possible," he added.
"There are returns flights planned in the coming days. And we will also continue to go after the heinous criminals and organised crime networks putting people's lives at risk."
More from Migrant Crisis
The government has asked the Navy for help and a former Royal Marine has been appointed "clandestine Channel threat commander" to try and cut down the numbers.
Image:Home Secretary Priti Patel
Home Secretary Priti Patel vowed last year that the crossings would have become an "infrequent phenomenon" by now.
She has since called the figures "appalling" and levelled blame at her counterparts on the continent as she revealed the UK and French governments were locked in a row over the interpretation of maritime law.
The Sunday Telegraph reported France wants £30m to cover the costs of increased policing to stop the migrants crossing into English waters.
It also claims ministers are looking into chartering spy planes and drones.
Migrants beach landing 'was a mad show'
Bridget Chapman, spokeswoman for the Kent Refugee Action Network, said the government's handling of the situation is "increasingly chaotic".
"Nothing they have done so far has worked," she said.
"They have spent millions of pounds of taxpayers' money fortifying the port at Calais, have increased patrols in the Channel, and are now escalating a failing strategy by calling in the military to deal with a humanitarian situation.
"Priti Patel has said that she wants to make the Channel 'unviable' for people attempting to cross. But what is unviable is her approach."
British family find migrants in their car roof box
A government source told Sky News earlier this weekend that a "passive blockade" in the Channel was being considered.
Natalie Elphicke, the Conservative MP for Dover, is calling for migrants to be returned to France irrespective of whether they are picked up in British or French waters.
Kent County Council has said that 400 migrant children have been taken into its care this year, including 60 in the first week of August and 23 on Friday alone.
A political row between the UK and France over migrant crossings in the Channel has intensified, with Britain's immigration minister saying the "unacceptable" increase in numbers "simply cannot go on".
Chris Philp is due to meet his French counterpart in Paris on Tuesday to discuss stronger measures, including "interceptions and returns".
It comes as 151 migrants reached the UK on Saturday - taking the total since 1 January to 4,100.
While southeast England basked in roasting sunshine on Sunday, a dozen people arrived in Dover after making the dangerous crossing.
Advertisement
Migrants met by Border Force in Kent
Mr Philp said the UK government wants to make the route across the Channel "completely unviable" so migrants "have no incentive to come to northern France or attempt the crossing in the first place".
"We intend to return as many migrants who have arrived as possible," he added.
More from Migrant Crisis
"There are returns flights planned in the coming days. And we will also continue to go after the heinous criminals and organised crime networks putting people's lives at risk."
The government has asked the Navy for help and a former Royal Marine has been appointed "clandestine Channel threat commander" to try and cut down the numbers.
Image:Home Secretary Priti Patel
Home Secretary Priti Patel vowed last year that the crossings would have become an "infrequent phenomenon" by now.
She has since called the figures "appalling" and levelled blame at her counterparts on the continent as she revealed the UK and French governments were locked in a row over the interpretation of maritime law.
The Sunday Telegraph reported France wants £30m to cover the costs of increased policing to stop the migrants crossing into English waters.
It also claims ministers are looking into chartering spy planes and drones.
Migrants beach landing 'was a mad show'
Bridget Chapman, spokeswoman for the Kent Refugee Action Network, said the government's handling of the situation is "increasingly chaotic".
"Nothing they have done so far has worked," she said.
"They have spent millions of pounds of taxpayers' money fortifying the port at Calais, have increased patrols in the Channel, and are now escalating a failing strategy by calling in the military to deal with a humanitarian situation.
"Priti Patel has said that she wants to make the Channel 'unviable' for people attempting to cross. But what is unviable is her approach."
British family find migrants in their car roof box
A government source told Sky News earlier this weekend that a "passive blockade" in the Channel was being considered.
Natalie Elphicke, the Conservative MP for Dover, is calling for migrants to be returned to France irrespective of whether they are picked up in British or French waters.
Kent County Council has said that 400 migrant children have been taken into its care this year, including 60 in the first week of August and 23 on Friday alone.
A WOMAN has died after a collision between a jet ski and a boat in North Wales.
Emergency services were called to the incident in the area of Menai Bridge, Anglesey, at around 7.30pm on Saturday.
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Emergency services were called to the incident near Menai Bridge, picturedCredit: AFP - Getty
A spokesman for North Wales Police said: "Officers received a call from the Welsh Ambulance NHS Trust at 7.29pm yesterday.
"One female casualty was transferred to hospital following the incident but has sadly since passed away."
The tragedy unfloded as a heatwave continues to sweep across Britan.
Temperatures will remain above 30C next week as thousands flock to the nation's beaches.
Cops have turned around many due to “red alert” overcrowding fears.
The top temperature yesterday was recorded at Frittenden, Kent, as the mercury hit 34.5C.
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Bournemouth, Christchurch and Poole Council's beach check app showed 18 of its beaches on red alert, warning people to avoid the areas because safe social distancing was not possible.
And further west in Dorset, police were turning furious drivers away from Durdle Door as roads in Lulworth shut because of the volume of visitors.
Following the tragic death in Wales, officers appealed for any witnesses to the incident to contact North Wales Police on 101 quoting reference number Y114906.
