The sanctions were imposed over an angry outburst Jones made on his web show InfoWars against an attorney representing the relatives of victims of the Sandy Hook school shooting.
The court rejected Jones' claim that his comments were protected by free speech rights.
Eight families are suing Jones for defamation over his claim that the shooting was a hoax.
WATCH from 2018: More families of slain Sandy Hook children sue Alex Jones
More families of slain Sandy Hook children sue Alex Jones
The families say Jones' comments have tormented them and subjected them to harassment and death threats.
"While the nation recoiled at the terrible reality of the shooting at Sandy Hook Elementary School, Alex Jones saw an opportunity," said the families' attorney Josh Koskoff in 2018. "He went on a sustained attack that has lasted for years, accusing shattered family members of being actors, stating as fact that the shooting itself was a hoax and inciting others to act on these malicious lies."
A gunman killed 20 first-graders and six educators in December 2012 at the Newtown, Connecticut, school. The gunman, Adam Lanza, fatally shot his mother before driving to the school to carry out the massacre and then killed himself.
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