funny how no topic here can survive unless it either rants about the pokémon's design or some troller trying to break down the game... says a lot about the fanbase... well, this sucky, forum-located fanbase >:}
This is merely speculation on my part but my guess is that you Pet Eevee enough in Amie to make her so happy that Pink and Blue Sparkles start to appear around her and her eyes change temporarily. Once this happens, you leave Amie and Eevee evolves (Can be canceled as well)
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IMO it would be so lame if Pokemon Amie was the method for evolving Eevee into Sylveon. How is that any different than increasing Eevee's friendship to obtain Umbreon or Espeon? We don't need Pokemon Amie and friendship, it seems redundant.