170 former pupils at St William's school in Middlesbrough claim that they suffered appalling physical and sexual abuse within the home.
The diocese claimed the Roman Catholic De La Salle Order of Brothers, which it hired to run the school, should share the liability for the cases of abuse.
In a landmark ruling, a judge at the Supreme Court agreed that the liability did not solely lie with the diocese.
Dr Jim Whiston says that the decision made by the Supreme Court has vindicated the diocese's view.
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The operation took around three hours to complete, with the remains being disposed of by a specialist animal by-product company.