Sarah Palin will doubtless appeal the verdict against her, and will almost certainly lose. Under existing libel law, she could only prevail if she could show so-called "actual malice," which basically means that the Times knew that the statement about her was false (knowing falsehood), or at least published it with a "high degree of awareness of … probable falsity" or "entertain[ing] serious doubts as to the truth of his publication" (reckless disregard of the risk of falsehood). Mere negligence or even gross negligence isn't enough. The jury has resolved against Palin this factual question about the Times editors' mental state, and the Second Circuit Court of Appeals is highly unlikely to overturn such a factual ruling.
In principle, of course, Palin could then petition the Supreme Court, asking it to exercise its discretion to review the case, and perhaps reconsider whether the "actual malice" test really should apply—the Supreme Court could reverse that precedent, even though the Court of Appeals can't. (Technically, the precedent isn't New York Times v. Sullivan, which required the actual malice test for public officials, but follow-on cases, such as Curtis Publishing v. Butts, which extended that to so-called "public figures.")
But it seems to me extremely unlikely that the Court would agree to hear Palin's case, because it arises under New York law—and a recent New York statute adopts the "actual malice" test as a matter of state law, whether or not it applies as a matter of First Amendment law.
