No, you should not do that. Any time you encrypt a message with CBC you should use a new, random IV and send it together with the ciphertext. Don't encrypt the IV after usage.
The IV for CBC mode has to be unpredictable. That means you can't reuse an IV and you should not take any already used value from elsewhere. That was the problem of TLS with the BEAST attack, you maybe have heard of it. As soon as an attacker can make any statements about the IV there could be a serious security problem with your scheme.
IVs are also used to make the encryption more random - with your scheme, your IV only depends on your message, so the ciphertext is always the same when the message is the same. We mostly don't want that because it would show an attacker that we are sending the same message again.