What's this? A woman? Gasp! Women don't show up very often in Borges' short stories, so it's always worth mentioning when he decides to include a female character, however briefly.
So who is this particular lady? Julia de Weidenau is a character in Jaromir's play, The Enemies: she's the woman with whom both Baron Römerstadt and Jaroslav Kubin are in love. As it turns out, the baron and Jaroslav the same person, so this sort of makes sense: Julia-the-love-object is part of their shared identity.
But Julia kind-of-sort-of exists outside the realm of the play-within-the-story, too. When Jaromir is sitting on the woodpile, smoking a cigarette and living the last sixteen minutes of his life, he tries "to recall the woman Julia de Weidenau had symbolized" (9). So this Julia is based on a real person, huh? Apparently she made enough of an impression on Jaromir for him to include her in his play, but not enough for him to remember her in his dying moments.
Jaromir's forgetfulness of Lady Julia might be a commentary on Borges' general attitude toward women in his work. We can picture Borges saying: "Women? Yawn. I'm too busy writing about abstract concepts like time and infinity to think about women." To which we say: um, dude, your loss.