From Facebook Help Center:
How do I set up a free @facebook.com
email address? To set up a free
@facebook.com address, go to your
Messages view and click the "Claim
your Facebook email" link.
Your email address will match your
public username, for example:
Profile: facebook.com/username Email:
If you don’t have a username yet, you
can choose one when you create your
Once you set up your email address,
people can email you using any
traditional email system (e.g.,
Hotmail, Yahoo or Gmail), and the
emails will be delivered to your
Facebook Messages. When you send
messages to external email addresses,
the emails will be formatted to look
like Facebook messages, including your
name, your profile picture and your
This seems to be the official information from Facebook but it points to receive email from traditional email systems, not to sending from elsewhere but inside of the Facebook web.
I hope you find a solution for this.