The judge made the rulingafter reviewing evidence at a hearing in London.
Doctors caring for Archie say continued treatment is not in his best interests and should end.
Archie’s parents, Hollie Dance and Paul Battersbee, from Southend in Essex, disagree.
But Mr Justice Hayden has now ruled that treatment can lawfully end.
He said: “This court has to ask itself whether continuation of ventilation in this case is in Archie’s best interests.
“It is with the most profound regret, but on the most compelling of evidence, that I am driven to conclude that it is not.
“Accordingly, the court cannot authorise or declare lawful the continuation of this present treatment.
“It is obvious from the detail of the treatment that I have set out above that it is intrusive, burdensome and intensive. If there were even a possibility that it could achieve some improvement to Archie’s condition, it would be both proportionate and purposeful.
“Where, as here, the treatment is futile, it compromises Archie’s dignity, deprives him of his autonomy, and becomes wholly inimical to his welfare.
“It serves only to protract his death, whilst being unable to prolong his life.
“Having come to this conclusion, there emerges the prospect of an end to Archie’s life, which reverberates more closely with the way he lived in the past.
“Arrangements can be made… that afford Archie the opportunity for him to die in peaceful circumstances and in the embrace of the family he loved.”
Archie’s family are being supported by a campaign organisation called the Christian Legal Centre.
Chief executive Andrea Williams said after Mr Justice Hayden’s ruling: “This is another devastating blow for the family and for Archie. Sadly, however, this is what we have come to expect from the courts in end-of-life cases.
“What Archie’s case has shown is that systematic reform is needed to protect the vulnerable and their families in end-of-life matters.
“Parents of vulnerable and critically ill children are being put through the mill at the most traumatic moments in their lives when what they need is compassion, support and respect from the NHS and the legal system.”
A lawyer representing Archie’s parents indicated that they wanted to try to challenge Mr Justice Hayden’s ruling in the Court of Appeal.
Family handout photo of Archie Battersbee
Hollie Dance
Archie Battersbee's mum, Hollie Dance
James Manning