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You may leave without an American passport. The United States does not have a government exit procedure, so at airport check-in, you would be showing whichever passport would be more convenient for your country of destination. If you were returning to the UK, for example, your UK passport would be better (although since US citizens do not need advance visa for UK, not that much better). Supposing you were a dual USA-Ruritania citizen and Ruritania requires advance visas for Americans, the advantage of the Ruritanian passport would be obvious.
You must, however, use your replacement USA passport when you next come into the United States.