Sending message body to email with PrivateMsg module
If you navigate to admin/config/messaging/privatemsg, you will find the template of mail which  get send by system when somebody private message to message some user.

You can modify the template. If you navigate to tokens below, you will find several tokens, one is - [privatemsg_message:original:body]. You can insert it in above template to send it in mail according to your need.

I think you need Token module for all this &amp; I hope you must have this installed in your instance.