People that discuss money during a resignation meeting often find that such personal information spreads like wildfire. Keep a level head. This is an emotional time. Be assured that your boss will have an emotional response. Let your boss know that in no uncertain terms, you will NOT consider a counteroffer.
Counters are bad news.
They are short term answers. People that accept counters destroy their loyalty and their career path within the company. The best thing that can be said about counters is that they are very flattering. Do not ever consider a counteroffer. Tell your two closest friends at work about your plans the night before you are scheduled to give notice and ask them to be confidential until you have met with your boss.