Hi Teo,
"Giving birth to". Hmm. I think theremust be another good way of saying this but I can't find now.
If you don't now the sex of the baby, use "baby", if you know choose "a boy" or" a girl" not "a daughte"r or "a son".
"Did you hear that Linda give birth to a girl?"
"Oh, really. I am happy for her. She has a daughter now!".