My iPhone 5 is having a strange problem that affects phone calls only. (And yes, I've tried rebooting.)

Regular Phone Call
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When I call someone, or someone calls me (whether via iPhone or non-iPhone), the call seems to connect, but there is **no audio** for either participant. They can't hear me, and I can't hear them.

The result is the same whether using speakerphone or not.

Microphone & Speaker OK?
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The microphone works fine -- I can make Facetime calls (either audio or video), I can have it convert my speech into text, etc.

The speaker you hold up to your ear works fine -- I can hear just fine on Facetime calls.

The speaker you use to broadcast sound to everyone nearby works fine, too -- I can play music and watch videos.

Network Behavior OK?
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The phone's ability to connect to the cellular network seems normal, too. I can make and receive Facetime calls via the cellular network, connect to the internet, and send/receive texts (for both iPhones and non-iPhones).

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What can I do to fix this problem? (Or at least diagnose it further...)

(After months of trying things, I was unable to fix the phone and simply bought a new one.)