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No, you don't. Most countries require you to have citizenship in another country before renouncing your citizenship. The US is not one of them. Americans have the right to make themselves stateless if they choose.

They will warn you that becoming stateless is a bad idea, but they won't stop you. Per the state department's website:

Persons who contemplate renunciation of U.S. nationality should be aware that they will experience a great deal of hardship unless they already possess a foreign nationality or are assured of acquiring another nationality shortly after completing their renunciation. In the absence of a second nationality, those individuals would become stateless. As stateless persons, they would not be entitled to the protection of any government. They might also find it difficult or impossible to travel as they would probably not be entitled to a passport from any country. Furthermore, a person who has renounced U.S. nationality will be required to apply for a visa to travel to the United States, just as other aliens do. If found ineligible for a visa, a renunciant could be barred from the United States. Renunciation of American nationality does not necessarily prevent a former national’s deportation from a foreign country to the United States as an alien.