The staff of the hospital in London, where terminally ill baby Charlie Gard is being treated for a rare genetic condition, have been harassed and even received death threats.
His parents have been battling in the courts for permission to take him to the United States for treatment, but Great Ormond Street says this would not help.
They have lost a succession of court cases to overturn the hospital's decision that it would be in the best interest of the child to be allowed to die with dignity.
The 11-month-old boy, who suffers from a rare genetic condition and has brain damage, is now the subject of an intense legal battle between his parents and medics over his treatment.
His parents, Chris Gard and Connie Yates, want to fly him to the U.S. for experimental treatment.
"The GOSH community has been subjected to a shocking and disgraceful tide of hostility and disturbance".
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"Great Ormond Street Hospital is in close contact with the Metropolitan Police and we will do everything possible to hold to account anybody who involved in this kind of deplorable behaviour".
Charlie's parents have lost all previous court cases, including one before the European Court of Human Rights, which were created to force the hospital to let them bring their son to the United States for an experimental treatment.
In a statement, Mr Gard said: "Without the excellent care of the doctors at GOSH [Great Ormond Street Hospital] our son would not even be alive and not a day goes by when we don't remember that". Some of the messages were allegedly so vile that they amounted to little more than death threats against doctors, nurses and staff. Staff have received abuse both in the street and on line. His case has attracted huge global attention, with USA president Donald Trump and the Pope intervening.
At a pre-court hearing on Friday, a lawyer for the hospital, Katie Gollop, said that the results of the child's latest brain scans make for "sad reading", the Associated Press reported.
He said the evidence must be new and relevant to the case.
Charlie has been examined by Dr. Michio Hirano, an American neurology expert from Columbia Medical Center in NY who has designed the proposed experimental treatment. The parents said at the hearing it was the first time they were being told about the latest results in the crucial test of Charlie's brain function.