Last month, a B.C. Supreme Court judge ordered the newspaper to pay Weaver $50,000 in damages last month.
Weaver filed the suit against the Post after it ran a series of columns published in 2009 and 2010 while he was a professor of climate science at the University of Victoria.
In her ruling, Justice Emily Burke said the articles, written by three different Post writers, defamed Weaver by unfairly questioning his credibility as a scientist. In addition to paying the monetary damages, the court ordered the paper to publish a full retraction and scrub the columns from its website.
But in a landmark decision, Burke also found the paper is not liable for allegedly defamatory comments posted under the online versions of the stories.
The newspaper has filed a notice of appeal Friday and wants the ruling overturned.
Weaver's lawyer, Roger McConchie, said they are now planning a cross-appeal and seeking an increase in the damages, and will likely try again to hold the paper accountable for the reader comments that Weaver claims defamed him.