I'd give them two weeks' notice. That's the professional thing to do. You don't have to tell them the pay is better. You don't have to give a specific reason at all, but you should say something. If they do tell you to leave that day, you may be able to move up your start date with the other company.
It sounds like you already have an offer letter in hand from the other company, but if you don't, get one.
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