Sorry: not true, so far as we understand.
If you believe quantum mechanics, those two pieces are in a certain sense still a single object ... even if the two pieces are on opposite sides of the room, or in separate towns many kilometres apart. It's called entanglement, a.k.a. "spooky action at a distance", a.k.a. "weird quantum s***".
I mention this because our paper on quantum entanglement at the Belle experiment has been published in Physical Review Letters. The theory does just fine at predicting our data --- that's not unexpected, since it's done just fine on all the data it's been confronted with. The real interest in this kind of measurement is to see if one can go beyond testing quantum predictions, and test entanglement itself: to show that entanglement is just-a-fact-about-how-the-world-is-put-together which we'll always be stuck with, even if we eventually improve on quantum mechanics in some way.
The gold standard for proving entanglement is a theorem by the late John Bell (no relation): our experiment couldn't meet this standard, even if our equipment were perfect (for rather technical reasons). What we can do is put other specific models --- other ways of explaining the data that don't involve entanglement --- to the test. The ones we have been able to try, fail; quantum mechanics succeeds. So entanglement wins this round, yet again, but some alternatives still live to fight another day ...
Here ends the lesson. It's not usually my aim to post such pedagogical material on this site, but there is no end to the flaky silliness on these topics doing the rounds in popular culture, so I feel some kind of duty to fly the flag when I've been a part of the work. What the bleep do we know? Um, well, quite a bit actually.
Gravitational and electromagnetic forces act "at a distance" but their influence is not instantaneous: it's bound by the speed of light. For everyday purposes that's so fast that the influence might as well be instantaneous, but a lot hangs on the distinction. You can think of it this way: it's the gravitational and electromagnetic fields right where you are ("touching" you) that affect you, and they take time to catch up on what's going on elsewhere, the same as you do. These forces are still local in this sense.
The "spooky" part about quantum entanglement is that the connection between the parts of an entangled system works without any regard to distance whatsoever --- with no speed limit --- yet it turns out that you still can't use the thing to send a signal faster than the speed of light. Put like that, it seems somewhat contrived, and this is one of the things behind the intuition that it's our assumption of separability that's the problem, not the assumption of locality: it's not that relativity doesn't describe spacetime, it's that things really can't be divided up into "parts" the way we tend to think they can.
 It's also publicly available on the arXiv preprint server as quant-ph/0702267.
 Like many people (physicists included) I have my doubts about quantum mechanics: I suspect that there's something more going on. However, I also suspect that the "something more" will still leave us stuck with entanglement: that the weirdness is real.