You don't even need the controller.
> Which Chromecast devices are compatible with Stadia?
> For launch in November, only the Google Chromecast Ultra will be compatible with Stadia, and you will also need a Stadia Controller and access to Stadia to play with Chromecast. Access to Stadia in November is only available through the Founder's Edition or through a Stadia Buddy Pass. Stay tuned for future announcements regarding additional device compatibility.