It's not supported right now. Discover stated they will be adding the service soon when it was announced last month. Keep this in mind though, "soon" is relative...
I knew from the first rumors of ApplePay that Discover would be the odd one out. Which sucks, because my only credit card is with them, and my debit card is with a community credit union. So looks like I won't be using ApplePay for a while.
I do wonder, though, when Discover eventually DOES support ApplePay, will Apple just... integrate them whenever they can? Or will we have to wait for another Apple announcement/keynote where they can use it as a bullet point?