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The discovery of the bodies - 38 adults and one teenager - was made early on Wednesday (local time) after emergency services were alerted to people in a truck container on an industrial site in Grays, about 32km east of central London.
Police said the trailer had arrived at nearby docks having travelled from Zeebrugge in Belgium and the bodies were found just over an hour later.
The red cab unit of the truck was believed to have originated in Ireland. It had "Ireland" emblazoned on the windscreen along with the message "The Ultimate Dream". The driver, a 25-year-old man from Northern Ireland, remained in custody.
Prime Minister Boris Johnson said he was appalled by the news and was receiving regular updates about the investigation which was focused on human trafficking.
"We know that this trade is going on - all such traders in human beings should be hunted down and brought to justice," he said.
All those in the container were pronounced dead at the scene after the emergency services were called to the Waterglade Industrial Park, not far from docks on the River Thames.
Police said the trailer had travelled from Belgium to Purfleet and landed shortly after 12.30am on Wednesday. It left the port on the truck at about 1.05am and ambulance services notified police about the discovery of the bodies at 1.40am.
Originally it was thought both parts of the vehicle had entered Britain at Holyhead in North Wales on Saturday and to have originally started its journey in Bulgaria.
The Bulgarian Foreign Ministry said while the vehicle was registered in Bulgaria by a company owned by an Irish woman on June 19, 2017, it had left the next day and never returned.
Bulgarian Prime Minister Boyko Borissov said his country had no other connection with the deaths.
'Desparate and dangerous situation'
Prime Minister Leo Varadkar said Irish authorities would carry out any investigations necessary if it was established that the truck had passed through Ireland.
Police officers in forensic suits spent the day inspecting the large white container on the truck next to warehouses and had sealed off much of the surrounding area of the industrial site with large green barriers as they carried out their investigation.
The truck was later driven away to a secure location at nearby Tilbury Docks so the bodies could be recovered.
"At this stage, we have not identified where the victims are from or their identities and we anticipate this could be a lengthy process," Essex Police Deputy Chief Constable Pippa Mills told reporters.
"I appreciate how much attention this incident will continue to attract and the public and media appetite to understand what's happened. We also need to understand what has happened," she added.
For years, illegal immigrants have attempted to reach Britain stowed away in the back of trucks, often seeking to reach the United Kingdom from the European mainland.
Separately on Wednesday, police in the neighbouring county of Kent found nine people alive in a truck on the M20 motorway, heading towards London, Sky News reported.
In Britain's biggest illegal immigrant tragedy, in 2000, customs officials found the bodies of 58 Chinese people crammed into a tomato truck at the southern port of Dover. It had begun its journey in Zeebrugge.
Jeremy Corbyn, the leader of the opposition Labour Party, said the latest deaths were an unbelievable human tragedy that needed answers.
"Can we just think for a moment of what it must have been like for those 39 people, obviously in a desperate and dangerous situation, for their lives to end, suffocated to death in a container," he said.