Moved Business Website, Keep a Goodbye Letter on Homepage, Point Search Engines to Different Domain

Let's say a certain small business is bought by a bigger company or closed down in favor of a larger business that can also service the customers of the closed business.

I don't want to just redirect the domain of the closed business to the new business' website out of the blue.

What is the best way of notifying the customers of the moving (e.g. using a farewell letter on the homepage), while at the same time pointing search engines to the new business immediately - without being seen as cloacking or black hat SEO?