I have just done the same thing. Luckily for me, my new employer backdated my hire date and my insurance started immediately. If they won't do that, I'd ask them to pick up your COBRA for the 90 days until you get the new.
If they won't do that, you have to pay the COBRA yourself. One day without coverage and the insurance company can whip out the pre-existing condition on you. Also make sure that the employer you are leaving gives you a certificate of credible coverage so that you know the EXACT date when your current coverage lapses so you can make sure you can prove that you were never without insurance. That is really the key. Stay covered no matter what or things could get expensive.