Yes, I realise this has already been suggested, but this was the one I thought straight away (and couldn't hurt its chances, if several people suggest it, and pile up reasons why:).
One version is there:
(Lot's of other Greek myths in there too, so search for Psyche to find the story)
Why this story?
It's not just a greek myth. It may have begun as one (or maybe it is even an older story adapted into Greek form), but it has since grown. For example, Beauty and the Beast is one version of the same story. Different details, but the heart is the same. There are other versions of this one too, I believe there is a Scandinavian version as well but couldn't find it right now. It is a story about loving the beauty inside, not just outside, and also about trust. Psyche is made immortal because her soul is as beautiful as her face, the beast is made into a man only when he learns to love and is loved despite his appearance, and so on. These are themes that will touch people regardless of their cultural background or upbringing.
Also, when thinking of great love stories, what comes to mind? Rome and Juliet, Anna Karenina, Wuthering Heights? Lots of tragedies. Cupid and Psyche is not one. It is a story of getting through hard times and situations, a story about hope.