If, as you indicate, the business that is being offered to you operates entirely within your exempt purpose, you could accept it and operate the business without paying unrelated business income tax on its “profits” because you wouldn’t have unrelated business taxable income.  You would record a large gift from the donor, which could affect your public support percentage, but could probably be eliminated as an unusual grant in the unlikely event that it would otherwise cause you to be treated as a private foundation.  (See Ready Reference Page:  “Calculating Public Support”)

If the business is a corporation, you need to be aware that if substantially all of its assets are given away in a single transaction, Treasury Regulations considered it to be a sale of the business at fair market value for which an income tax is due on the gain.  That could be a significant chunk of the value.  You don’t want to be given the stock and then have to pay the tax to liquidate the business to operate it directly.  Likewise, the donor doesn’t want to give you substantially all of the assets and have to pay the tax personally.  You should both consult with knowledgeable nonprofit tax lawyers to work out the best method to do a deal that would avoid significant income tax obligations.  There are lots of possibilities that you could consider.  100% donation may be the least tax-favorable one.