That's definitely illegal.
Maybe fraud and breach of contract if, in the full-time contract, it says "you affirm you do not have another full-time job that will interfere with this job".
Again, I don't do it because I can't handle the stress of two jobs, but I don't begrudge those who can
Additionally, it's a pretty obvious opening to a lawsuit in at least most jurisdictions for breach of fiduciary duty (under the duty of loyalty -- Google this + your own jurisdiction for more details on what it covers where you are) and breach of contract.
Now, the employer probably won't sue or prosecute in most circumstances, so you may get away with it. But that's some high-stakes roulette, especially since you're not going to get UI if you get fired for gross misconduct and you're not going to get a good recommendation.
IMO this is a no-brainer, the answer is yes, double-bill. The same applies if you get two remote jobs, you bill both.