I know you probably thought of this, but did you ever record the message someplace else or possibly have it playing while shooting a home video during a birthday or an anniversary? Probably a remote possibility, but you never know. Since you didn’t record over it, there might be a possibility of retrieving it. I really hope so! Please let us know what happens.
The much safer option is to do this with a third-party voice mail system like Google Voice (but the voicemail recordings are limited to 3 minutes). So it's better to have a paid voicemail service, use the Google Voice recording option for incoming calls outlined above, or spring for the paid recorder services like Recordator.

To set up your voicemail, press *98 from your home phone or call the retrieval number you received with your welcome letter. Voice prompts will guide you through the rest of the steps.
If your landline doesn’t have a way to save messages already built in, the best way to save that gorgeous message is with a recorder. But before you rush out to get it, don’t forget that most smartphones have a built in recorder. Try that first and see if it will pick up her voice loud and strong. It should save the message as an MP3 automatically, that you can send (via email or text) directly to your computer for safe keeping. If the quality is not great, then you can borrow a recorder or purchase one. We like the Olympus brand ourselves, because they save in MP3 and can upload to a computer wirelessly. We have the vtech phone/answering system indicated above. The outgoing message/greeting is of my young daughter’s voice and we would like to save it somehow. Am wondering if you have a process or can suggest how an outgoing/greeting message can be downloaded for safekeeping.