This is right in general but wrong on the specifics of this case. The Second Circuit Court of Appeals ruled on the original, federal trial. That court threw overturned the conviction entirely; it did not order a retrial.

The state prosecution was an entirely different matter, which is why they had to strain to pretend it wasn't double jeopardy. The "acts" were the same but they had decided to classify them as different, lesser crimes. In fact he had already served more time than he could have been sentenced under those, rendering the whole proceeding nothing more than a farcical publicity stunt.