As others have noted, US law says that if you have over half the bill, it's worth the full value, under half is worth nothing. I presume if it is very close to half, if even careful measurements show that you have 50.5%, you'll have difficulty cashing it in, precisely because the government and the banking system aren't going to allow themselves to be easily fooled by someone cutting bills in half and then trying to redeem both halves.
I've seen several comments on here about how you'd explain to the bank how so many bills were cut in half. What if you just told them the truth? Not the part about killing someone, of course, but tell them that you made a deal, neither of you wanted to bother with complex contracts and having to go to court if the other side didn't pay up, so your buddy cut all the bills in half, etc. As you now have both halves and they clearly have the same serial number, this no real evidence of fraud. Okay, this is technically illegal -- 18 US Code Section 333, "Whoever mutilates, CUTS, defaces, disfigures, or perforates, or unites or cements together, or does any other thing to any bank bill, draft, note, or other evidence of debt issued by any national banking association, or Federal Reserve bank, or the Federal Reserve System, with intent to render such bank bill, draft, note, or other evidence of debt unfit to be reissued, shall be fined under this title or imprisoned not more than six months, or both." But you didn't do it, the other guy did. I presume the point of this law is to say that you can't get a hold of currency belonging to someone else and mutilate it so as to make it worthless. As he's now given you both halves, I doubt anyone would bother to track him down and prosecute him.
Just BTW, while checking up on the details of the law, I stumbled across 18 USC 336, which says that it's illegal to write a check for less than $1, with penalties of 6 months in prison. I just got a check from AT&T for 15 cents for one of those class action suits where the lawyers get $100 million and the victims get 15 cents each. Apparently that was illegal.