The story is that of Pyramus and Thisbe, from Ovid's _Metamorphoses_. You have already covered almost everything he tells us in your question, but there are a couple of extra details: They were living in Babylon; they were meeting under a mulberry tree), and it's fruit took colour from Pyramus' blood (the transformation that makes it fit in the Metamorphoses), and their ashes are kept in the same urn.
The story was retold several times; Shakespeare himself used it as the play that the "rude mechanicals" enacted in _A Midsummer Night's Dream_. It was also used by Chaucer and Boccaccio. If you want to read Ovid's account, Wikisource has several translations that are in public domain.