The man suffered serious neck injuries after falling from a low roof at Melbury Lodge, Royal Hampshire County Hospital, in December 2015.
The Care Quality Commission (CQC) brought the prosecution against Southern Health, who run the Winchester hospital.
District Judge Philip Gillibrand adjourned for sentencing in October.
Paul Spencer, representing Southern Health, entered a guilty plea at Basingstoke Magistrates' Court to the offence of failing to provide safe care and treatment and failing to assess risk to patients at the unit.
The court heard the patient, named in court as "AB", had already climbed onto the roof during an earlier stay at the unit.
Paul Greaney QC, prosecuting for the CQC, said Southern Health had failed to take action to prevent patients from gaining access to the low rooftop between 2010 and 2016.
"The trust was aware of that dangerous state of affairs but did nothing sufficient to stop it until late April 2016," he added.
Judge Gillibrand said Southern Health faced an unlimited fine for the offence.
He added: "It's clearly a case of considerable public interest and concern."
Southern Health has previously been criticised over its failure to properly investigate the deaths of hundreds of patients in its care between 2011 and 2015.
The Melbury Lodge unit at the hospital treats people with severe mental health problems.
Southern Health NHS Foundation Trust has admitted failings over the fall of a mental health patient from a roof.