I just decided to release this to the public. It uses your credentials to make a separate connection to Spotify while listening to media broadcast events sent from the android Spotify client. When one is received it makes a request via the connection for the required metadata and track data. It is then forwarded to your Chromecast to be played. There is a slight delay and a few bugs here and there but overall should be stable enough for a public release. Keep in mind it has not been tested externally until now. You must have a Spotify premium account and you must enable "Device Broadcast Status" in Spotify!
I have no affiliation with Spotify. I think they are a great company who provide a great service but I do not approve of their choice to exclude support for Chromecast. So here it is.
You can find the application in the Google Play store under the name Spoticast.
You can find a post with information FAQ, issues and other discussion on reddit in /r/Chromecast