Thread title says it all.
After I posted this, I finally found it. Apparently it means you deal half damage. Reading FTW!
An interesting concept which Might & Magic VI had twelve years ago (before 3rd Ed). Wonder how they thought of that one? 
Yup, a weakened character deals half damage on all attacks until he is no longer weakened. Note that this includes not just melee attacks, but ranged attacks and spells. (No, it doesn’t make much sense that your wizard, while weakened, only does half damage with Magic Missile, but there it is…)
Which, of course, is only natural given that 4e makes no other distinction between what a warrior does and what a wizard does, either.
This one got house-ruled out fast for us for those reasons. A pre-loaded crossbow suddenly becomes half-effective? While somehow still maintaining its original range? A hurled axe flies just as far and fast but doesn’t hurt as much when it hits? The boulder dropped on you by a storm giant doesn’t land as hard?
In 4e, it is far better to come up with a better explanation for WHY attacks are doing half damage than to change the rule. They put a lot of work into balancing the game, but that balance is precarious, and too much screwing with it brings the whole house of cards down very quickly. Screwing with the rules for aesthetic reasons is the fastest way to make the game not work properly as a game.
It’s like how anything can be knocked prone, even things like slime where that doesn’t make sense. You don’t make things immune by fiat, you just retheme the flavor into something appropriate.
So, back to weakened. The archer’s shaky hands make the bolt fly slightly off, and the target has time to turn so that a serious wound becomes a flesh wound instead. The axe thrower has to put more effort into throwing the axe at all so he can’t judge the spin as well, so although it still hits the blade doesn’t sink as deep. The storm giant, instead of throwing the boulder directly at your head, rolls it down the cliff and strikes a more glancing blow.
Think of it as a creative exercise. God knows that 4e NEEDS more creative exercises to keep it fresh.
And setting the rules by game balance reasons, without regard for aesthetics, is the fastest way to make the game not work properly as role-playing.
One might argue that this has already been done, and trying to half-arse it back the other way with house rules is only going to result in a game the works as neither. Suggestion: If you are trying to play a roleplaying intensive or simulationist game, pick a different game system, there are lots of them.
Relevant article by Yours Truly here (scroll down to Game Balance Redux)
Which is precisely why I’m currently hand-rolling my own simplified version of D&D.
The 4th edition rules are over-tweaked and soulless.