He has to persuade more than a majority of the jury.
It depends. Since this is a federal trial, it is possible that a verdict would not have to be unanimous, though both sides would usually have to agree. But even then, getting 6 people (the number of jurors on a typical jury in a federal civil case) to agree that it is more likely than not that he defamed her is not that high of a bar to clear, especially in New York.