Monroe, a food blogger who also campaigns over poverty issues, sued Hopkins over tweets they said caused “serious harm” to their reputation.
After Mr Justice Warby announced his findings on Friday afternoon, Monroe who identifies outside the binary construct of gender and prefers the gender-neutral pronoun ‘they’ and title ‘Mx’, tweeted of their relief.
They wrote: “It’s taken 21 months but today the High Court ruled that Hopkins statements to/about me were defamatory.
“I sued her for libel, and I won...To everyone who told me I couldn’t, wouldn’t, shouldn’t - I could, I would, I did.”
The judge also ordered Hopkins to pay an initial £107,000 towards Monroe’s legal costs within 28 days. The final costs figure has yet to be assessed, but some legal experts put it above £300,000.
He ruled the tweets had caused Monroe “real and substantial distress” and they were entitled to “fair and reasonable compensation.”
Monroe, who did attend the hearing, asked the London court to find that they had been “defamed” by the former Apprentice contestant.
At the heart of the action is a posting on Twitter in May 2015.
Monroe says the tweet meant they had either vandalised a war memorial, and “thereby desecrated the memory of those who fought for their freedom and had committed a criminal act”, or that they “condoned or approved” of the criminal vandalisation of a war memorial.
Hopkins, who didn’t attend Friday’s hearing, posted: “@MsJackMonroe scrawled on any memorials recently? Vandalised the memory of those who fought for your freedom. Grandma got any more medals?”
Monroe responded almost immediately: “I have NEVER ‘scrawled on a memorial’. Brother in the RAF. Dad was a Para in the Falklands. You’re a piece of shit.”
A Twitter debate had earlier erupted after a memorial to the women of the Second World War in Whitehall was daubed with the words “Fuck Tory scum” during an anti-austerity demonstration.
Monroe, 28, from Leigh-on-Sea, Essex, also claimed that a second tweet by Hopkins bore a “defamatory innuendo meaning” that they “approved or condoned the criminal vandalisation of the women’s war memorial in Whitehall during an anti-government protest”.
Their lawyer William Bennett told the judge: “The claimant’s primary case is that by reason of the seriousness of the allegations and the scale of publication, serious harm to reputation has been caused.
“A widely published allegation that someone has either vandalised a war memorial or approved of such an act will inevitably cause serious damage to reputation.”
Mr Bennett said “the libel was a particular affront... because part of their identity is as a member of a family closely involved with the armed forces”.
Even after Hopkins deleted the first tweet, “she did not apologise or retract the allegation even though she knew it was false”, he said.
Jonathan Price, for Hopkins, told the judge her case was that “this relatively trivial dispute arose and was resolved on Twitter in a period of several hours”.
He argued that “no lasting harm, and certainly no serious harm”, to Monroe’s reputation resulted from it.
Mr Price said her case was that “these proceedings are an unnecessary and disproportionate epilogue to the parties’ otherwise forgotten Twitter row”.
Hopkins had “mistakenly” used Monroe’s Twitter handle instead of that of another columnist who had tweeted about the memorial incident.
Monroe told the court: “These proceedings have been a nightmare. It has been an 18-month, unproductive, devastating nightmare.
“I have offered several times through my lawyer to settle these proceedings outside court. This is the last thing that I wanted to be doing.”
Monroe’s lawyer, Mark Lewis, a partner at Seddons Solicitors said: “Jack Monroe has finally been vindicated in full from the libellous and wholly false accusation by Katie Hopkins that she had supported the vandalisation of a war memorial. Jack Monroe never did, and coming from a proud military family, never would.
“Despite pointing this out to Katie Hopkins within minutes of her first tweet wrongly accusing Jack, Hopkins did not apologise. Rather the self-styled ‘rent-a-gob’ defiantly posted another defamatory tweet. Despite repeated opportunities to back down, Hopkins obstinately refused to apologise, instead conducting her defence by slinging as much mud as she could to hide the fact that she had made this false allegation.
“The price of not saying sorry has been very high. Hopkins has had to pay out of her own pocket a six figure sum in damages and costs for a tweet that should have been deleted within minutes as soon as she was told it was wrong. On this occasion, the cost of renting that gob was particularly high.
“Hopkins claimed that Twitter was just the wild west where anything goes. The Judge has shown that there is no such thing as a Twitter outlaw.”