The 2nd Circuit Appeals Court stayed the order:
Gawker appealed the injunction, and Florida’s Second District Court of Appeal issued an immediate stay, which prevented the injunction from going into effect. Eight months later, the appeals court issued a scathing opinion that overturned Campbell’s order on the grounds that the video was newsworthy and Gawker's publication of it was protected by the First Amendment.
Armed with the appeals court decision, Gawker went back to Campbell and asked her to dismiss the case, since the appeals court had just ruled that publishing the video was protected by the First Amendment. Campbell refused.
http://www.politico.com/media/story/2016/03/jury-awards-hulk...