Apple Music is now available on the Amazon Echo:
To set up Apple Music on the Echo, open the Alexa iOS app and go to the Music section of the app’s Settings. You won’t see Apple Music listed as an existing service, so tap the ‘Link New Service’ button, which will list Apple Music along with several other services.
Once you’ve logged in with your Apple Music credentials, the Apple Music Skill is enabled, which allows you to say things like ‘Alexa, play my New Music Mix on Apple Music.’ You can separately set Apple Music as your default music service, which lets you use Alexa to request music without specifying ‘on Apple Music.’
I’ve been playing with it all weekend and it works great. It doesn’t (currently) have access to your entire iCloud Music Library, so things in your personal collection aren’t available, but the millions of songs on Apple Music are all available, as are your custom playlists.