An Essex Police investigation into an alleged people-smuggling ring linked to the deaths led to charges against five men, including lorry driver Maurice Robinson.
During a virtual hearing at the Old Bailey, 25-year-old Robinson admitted 39 counts of manslaughter on or before October 24 last year.
Robinson, of Craigavon in Northern Ireland, had previously pleaded guilty to conspiracy to assist unlawful immigration of non-European Union citizens between May 1 2018 and October 24 2019.
The bodies of eight females and 31 males were discovered in the container attached to his Scania cab in an industrial park in Grays, Essex, early on October 23 last year.
They were later identified as coming from various provinces of Vietnam, with the youngest being two boys aged 15.
The lorry arrived on a ferry from Zeebrugge in Belgium.
Five men charged following an investigation by Essex Police appeared for a virtual hearing at the Old Bailey before Mr Justice Sweeney.
Robinson had previously pleaded guilty to conspiracy to assist unlawful immigration and acquiring criminal property.
He denied a further charge of transferring criminal property.
Robinson appeared at court via video link alongside four other co-defendants.
British Romanian Gheorghe Nica, 43, of Mimosa Close in Langdon Hills, denied 39 counts of manslaughter.
He also denied one count of conspiracy to assist unlawful immigration between May 1 2018 and October 24 2019.
Romanian national Alexandru-Ovidiu Hanga, 27, of Tilbury, denied a charge of conspiracy to assist unlawful immigration.
Christopher Kennedy, 23, of Darkley, Co Armagh, Northern Ireland, has previously denied conspiracy to assist unlawful immigration
Valentin Calota, 37, of Birmingham, was not asked to enter a plea to the charge of conspiring to assist unlawful immigration.
Prosecutor William Emlyn Jones QC said a human trafficking conspiracy charge was being dropped in relation to Kennedy and Robinson.
He asked for three weeks to decide whether to proceed with a trial against Robinson on the outstanding charge he faced.
The other defendants face a trial at the Old Bailey lasting up to eight weeks from October 5.
The hearing was conducted virtually with most lawyers and court reporters attending by Skype.
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