I had setup a distribution list where one user had the rights to send as / send on behalf configured. I knew that we needed to show the From field by clicking on the appropriate menu item.
However, to change the from e-mail address to the distribution list's e-mail address was not as straightforward as I thought. I figured that the user would be able to click the down arrow next to "From" and select the other e-mail address, but no other address shows up.
What you have to do is right click on your e-mail address and delete it. Then you can start typing the e-mail address you want to use and "Search Directory". You'll see something like this in which you can select which e-mail to send as.