Simply search for "Apple Music" in the skills section of the Alexa app and hit "Enable to use" to link your Apple Music account. You will be asked to allow Alexa to access Apple Music and can even designate Apple Music as your default music player so you don't have to constantly specify "Apple Music" in your command.
It was easy to set up and worked well for me during a cursory test. My Amazon Echo Dot did have to go through a short software update first, but that, along with enabling the Apple Music account, was all it took.
Unlike the Apple HomePod, which only supports Apple Music natively, Amazon speakers work with Spotify, Pandora, Amazon Prime Music, Amazon Music Unlimited, iHeartRadio, TuneIn and SiriusXM -- in addition to Apple Music.
Apple Music has, making it the . Spotify still holds the lead worldwide.
CNET Smart Home
reading•Apple Music now works on Amazon Alexa smart speakers
Jan 22•Pay your respects at the smart-home graveyard
Jan 22•Amazon just made it a little bit easier for Alexa Skills to make money
Jan 22•Philips Hue, Ring, Nanoleaf, GE: Tons of new smart lights coming in 2019
Jan 20•Amazon's Echo Wall Clock puts your Alexa timers in full view