The BBC Trust said the BBC failed to ensure the students were aware of the risks involved in the trip in order to give their consent, which was a "serious failing". And it concluded there had been "unfair treatment" of the LSE by being linked to the BBC's investigation.
For the programme, North Korea Undercover, reporter John Sweeney spent eight days in the country, joining an organised tour and pretending to be part of the student group.
When the film came to light it was condemned by officials from the LSE, who asked for the programme to be pulled, but it was screened in April 2013 which led to a complaint from the father of one of the students.
The Trust said the BBC "failed to consider a number of important issues and risks, and failed to deal with them appropriately".
BBC News said it accepted the Trust's decision and said it had apologised to the LSE and the student whose father had complained. But it said at the time, it believed they were being treated "fairly".