While this works, note that it isn't officially documented with the API, so it may change and stop working in the future. I'll send this along as a feature request for an officially supported way to retrieve the URL via the API.
We love to learn from the educators who use Dropbox. Whether you teach kids, teens, adults or a combination of all three, we want to know what apps and integrations you use with Dropbox to help with teaching. Which of the ones below is your favorite, or most used tool?