Yeah, I know. But if it is the case that they're wanting to hire someone immediately, I don't want to lose this opportunity because of it.
It's your responsibility to ask for the two week notice. If they don't agree with that, you can try to negotiate something with both parts. Maybe you can work half time in one and half time in the other one.
If they don't agree even with that, then you have to decide if it's "worthy" to drop your old job.
personally, I wouldn't want to "drop" my old job, it's an unprofessional thing to do, and you never know when an old contact from an old job can be useful.