You prepare for the layoffs quickly and quietly. Then one morning you do them. All of them. And then have a company meeting and you tell the people that that was it. No more layoffs. You're on the team. And then have a good story about how you're going to lead them to prosperity.
If you let the right people go, everyone will know. There's always fat in an organization and the people know where it is. The question is, do you know. If you cut badly you probably just blew your chance for recovery.
And you don't want to do what AOL is doing. Week after week cutting off limbs. So that everyone inside the company is thinking they're next. So the smart ones look for new jobs in earnest. By the time you get around to looking for the next place to cut, all you'll find are the ones who didn't get offers elsewhere. You just lost.