pretty much what's been said so far:
1) get in touch with FL and ask them what you'd be doing - do they need a research assistant who'd be doing actual research or a water boy
2) talk to the state school program, be honest with them, don't play mind games and try to be too smart with them 'cause it will probably come to bite you in the behind later on.
Just tell them how things are and ask if there is any chance to push back with 10 weeks or not and if, should Fermi lab extend your stay with them, you could stay at FL ... who knows, the state school might not be in such dire straits for an RA and maybe in a position to wait 10 weeks to see how your situation develops and only then hire someone else in case you stay at FermiLab.
Unlikely, but won't hurt to ask
Of course, you should tell the state school that FL is offering you research in the area you intend to pursue in grad school, so they would probably understand you position on this one ... and they know well that an RA at Fermilab is a dream for any aspiring grad student.
3) As gray mentioned, there is a good chance that Fermilab is offering you just 10 weeks as a try-out period, so bust your but, bring bar endeavors to a minimum and do your best to impress them - you do that and I bet they will "miraculously" come up with extra funds to extend their stay with them.
4) even if FL end up NOT keeping you after the 10 weeks are over, you can always find something else - even then, however, you'd have a stint at Fermi on your grad school application, which in all likely hood would beat any state-school RA especially one that is not in your intended area of research.
In short, whatever happens after those 10 weeks, going with FermiLab would be worth it - it is that good of an opportunity and you will most likely hate yourself if you were not to take it ... at the same time, there are lots and lots of state schools out there, not exactly the chance of a lifetime
First things first, though - ask the FL group what exactly they need you for.
Good luck and congrats - these are sweet issues that you are having